Stabilization of perturbed systems via linear optimal regulator

Soroka and Shaked have shown that a linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system can lead to fast unstable closed-loop modes. It is shown there that the stability properties can be improved if possible nonminimum phase behavior is allowed for in modeling the system prior to the controller design.

In this paper, a discrete state feedback Linear Quadratic Regulator (LQR) for event-triggered control is presented. To ensure zero steady state error in the case of such controllers, one normally extends the states with an integral action. Instead of using integral action, the idea is to estimate the disturbance causing the steady state error and use this to extend the states. A Lyapunov-based event triggering function is proposed. Practical results using a gyroscope actuator are presented and compared to a classical time-triggered controller. The obtained results demonstrate the simplicity and efficiency of the proposed approach.

This paper concerns the problems of robust stability analysis and stabilization for uncertain linear system with time-varying delay. In order to obtain a robust stability criterion with less conservative for uncertain linear delay system, a quadratic integral inequality is improved by matrix decomposition. Further, two different kinds of controller, i.e., linear memoryless state feedback controller and linear state feedback controller with state delay, are derived in terms of the robust stability criteria. Two numeral examples are provided to demonstrate the validity of the proposed methods.

This paper presents a technique for designing generating units linear optimal regulators with extended stability region around nominal operating points. The technique uses a stability robustness criterion for the selection of the relative weights between the control effort and the system state regulation In the linear regulator quadratic performance index. The resulting regulator is stability robust in the sense of tolerating a specified set of unit parameter variations. By proper specification of the parameter variations, a robust linear optimal regulator Is obtained which guarantees unit stability in an extended region of operation around a given operating point. Example results are presented to substantiate the proposed technique by application to the design of a linear optimal regulator for a generating unit excitation system.

This paper investigates the infinite-horizon linear quadratic regulator (LQR) problem of continuous-time LTI systems with random gains imposed on the input channels. The main novelty of this work originates from the point of view that in networked control, designing the channels and controller jointly often leads to an easier problem and meanwhile achieves better performance than designing them separately. Specifically, we formulate the LQR problem with random input gains as a channel/controller co-design problem. Such co-design can be realized by the twist of channel resource allocation, i.e., the channel capacities can be allocated among the input channels subject to an overall capacity constraint. The LQR problem is shown to be attainable under such co-design if and only if a modified algebraic Riccati equation (MARE) has a mean-square (MS) stabilizing solution. The optimal controller is given by a linear state feedback associated with the MS stabilizing solution. Moreover, a Newton's type iteration is developed for the computation of the MS stabilizing solution to the MARE.

It is well known that linear quadratic regulator can have excellent stability robustness. However, some authors have recommended caution, by producing examples where the linear quadratic regulator may suffer from poor robustness. Specifically, small changes in the parameters of the system can cause fast unstable closed-loop poles. However, by comparing the bounds on allowable perturbations it is shown that the stability robustness properties are preserved in a general sense. The high parameter sensitivity gives a warning to those designers who favour high gain controllers.

A global sliding mode controller (GSMC) is proposed for the missile electromechanical actuator (EA) servo system, where exists high uncertainties, such as parameter variations and external disturbances. By the design of an optimal integral switching function based on optimal linear quadratic regulator (LQR) theory, the initial state of system is set on the switching surface, and the optimal sliding mode motion is produced. The proposed GSMC is composed of an optimal linear state feedback controller (OLSFC), and a fuzzy nonlinear robust controller (FNRC), which can be designed respectively. The OLSFC, generated by the designed switching function, intends to minimise a quadratic performance index, and then improves the dynamic performance of system. Meanwhile, the FNRC employs a fuzzy decision maker (FDM), which estimates the upper bound of uncertainties as FNRC’s gain adaptively, and then makes GSMC robust and control input smooth. With the computer simulations on an EA experiment plant, it presents that the proposed scheme possesses good tracking precision, effective suppression against chattering at control input, and strong robustness against system uncertainties.

This paper proposes a polynomial chaos based numerical method providing an optimal controller for the linear-quadratic regulator (LQR) problem when the parameters in the formulation are uncertain, i.e., a controller minimizing the mean value of the LQR cost function obtained for a certain distribution of the uncertainties which is assumed to be known. The LQR problem is written as an optimality problem using Lagrange multipliers in an extended form associated with the polynomial chaos framework, and an iterative algorithm converges to the optimal answer. The algorithm is applied to a simple example for which the answer is already known. Polynomial chaos based methods have the advantage of being computationally much more efficient than Monte Carlo simulations. The Linear-Quadratic Regulator controller is not very well adapted to robust design, and the optimal controller does not guarantee a minimum performance or even stability for the worst case scenario. Stability robustness and performance robustness in the presence of uncertainties are therefore not guaranteed. However, this is a first step aimed at designing more judicious controllers if combined with other techniques in the future. The next logical step would be to extend this numerical method to H2 and then H-infinity problems.

For the longitudinal motion of air breathing hypersonic vehicles (AHFVs), a minimax linear quadratic regulator (LQR) control is synthesized considering seven uncertain aerodynamic and inertial parameters. The minimax control strategy works under the assumption that the uncertainties satisfy certain integral quadratic constraints, and provides robust stability as well as robust performance for the AHFV system with uncertainties. The design utilizes the effectiveness of the widely used feedback linearization technique with a special treatment of the non-vanishing nonlinear terms in the presence of uncertainties. The outputs to be controlled are the longitudinal velocity and altitude, and the control variables are the throttle setting and elevator deflection. The performance of the controller is evaluated at a particular cruise condition at Mach 15 with 100ft/sec and 2000ft commands for velocity offset and altitude respectively. Results for the proposed controller are also compared with results in the literature utilizing adaptive sliding mode and stochastic robustness based controllers , and the proposed minimax controller is demonstrated to be more robust than these approaches in the presence of uncertainties.

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.

Sit-to-stand (STS) movement is a complex functional task which requires coordination of movement, postural regulation and stability for successful execution. STS transfer consolidates the highly nonlinear musculoskeletal structure together with neural control and tactile system in human body. In this paper we propose a nonlinear control technique based on feedback linearization to emulate the control action of central nervous system in performing STS movement. We use 4-segments rigid body biomechanical model with 3 degrees-of-freedom built on average anatomical proportions. In this controlling scheme, we adopt the output feedback computing through physiologically relevant optimization based upon center of mass (COM) and ground reaction forces (GRF). Furthermore, the output feedback provides passive control action commands including a linear quadratic regulator (LQR) based function augmented with nonlinear function computed with feedback linearization. The reference trajectories generate active feed forward torques, in addition with passive torques to settle the motion profiles within human anatomical constraints. The simulation results show that feedback linearization in combination with LQR provides an optimal frame work for better results of biomechanical STS movement as compared to previous linear control design schemes.

A pole placement method was developed by modifying the symmetric root locus method of a class of linear quadratic regulators (LQR). It was successfully applied to the case of a coordinate measuring machine. The performance and robustness properties of this method were also examined by the same authors. The need for a slight modification of this method to include theoretical guarantees on the robust stability and performance was emphasized. In this paper, we present a method, originally developed by Megretski, as an suitable modification to include uncertainty description and saturation type nonlinearities. We also derive the discrete time results that makes the method suitable to implement in practical applications. The presence of nonlinearities like saturation or rate limiters in addition to parametric uncertainty complicates both the analysis and design procedure. This method is implemented on a 3D-coordinate measuring machine which exhibits nonminimum phase behavior. Significant improvements and good robust performance is exhibited over the traditional LQG method.

Coordinated control of multiple HVDC links using input–output exact linearization

Directional Steering Systems (DSS) have been developed in the oil and gas industry to achieve high well production when compared to conventional drilling systems. In this paper a dynamic modeling of a DSS using four DC motors was developed. A controller was designed using Feedback Linearization technique for the cancellation of non-linear dynamics and LQR for the optimal linear control. Simulation of the proposed 4-motors DSS is provided, and the performance of the Feedback Linearization Controller (FLC) - Linear Quadratic Regulator (LQR) based command and control in executing steering tasks is demonstrated.

In this paper, the inverse optimal approach is employed to design distributed cooperative control protocols for identical linear systems that guarantee consensus and global optimality with respect to a positive (semi-) definite quadric performance index. Cooperative control and pinning control problems are considered, where the communication graphs are assumed to be directed and have fixe topology. Simple sufficient conditions are established, which indicate that the global optimality is achieved using local distributed protocols which are designed by the linear quadric regulator (LQR) based optimal control method. Examples are given to show the effectiveness of the proposed methods.