Algebraic Ricatti Equation Research Articles

Third Order Observer Design for a Rotating Energy Harvesting System - A Brief Paper

In most nonlinear systems, not all the states are available for measurement. In such instances, an observer is utilized to estimate the unavailable states. Designing observers for nonlinear systems has received widespread attention owing to its utmost significance and intensive mathematical computations generally, an observer is used to calculate approximately the mechanical quantities from the identified electrical variables in an energy harvesting system. In the past, several estimations remained a challenge because of the rigorous computational techniques involved. In this work, we present a means of designing an observer by using the algebraic Ricatti equation.

Less conservative conditions for robust LQR-state-derivative controller design: an LMI approach

ABSTRACT This study proposes less conservative conditions for robust linear quadratic regulator controllers using state-derivative feedback (SDF). The algebraic Ricatti equation was formulated using the SDF, and its solution was obtained by linear matrix inequalities. SDF was chosen owing to the presence of accelerometers as sensors. Since accelerometers are the main sensors in mechanical systems, the proposed technique may be used to control/attenuate their vibrations/oscillations. Moreover, to formulate the less conservative conditions, some methods in the specialised literature were used, such as, for example, slack variables by Finler's Lemma. The paper also offers necessary and sufficient conditions for an arbitrary convex combination of square real matrices to be a nonsingular matrix, and thus an invertible one: must be nonsingular and all the real eigenvalues of must be positive. This result is important in the formulation of the proposed less conservative conditions since it was assumed that a given convex combination is nonsingular. A feasibility analysis demonstrates that the proposed conditions reduce the conservatism. Thereby, it is possible to stabilise a higher number of systems and to reduce the guaranteed cost. Furthermore, a practical implementation illustrated the application of the proposed conditions.

SDRE-Based Integral Sliding Mode Control for Wind Energy Conversion Systems

This paper proposes a novel integral sliding mode control (ISMC) scheme based on numerically solving a state-dependent Ricatti equation (SDRE), nonlinear feedback control for wind energy conversion systems (WECSs) with permanent magnet synchronous generators (PMSGs). Unlike the conventional ISMC, the proposed control system is designed with nonlinear near optimal feedback control part to take into account nonlinearities of the WECSs. The Taylor series are used to approximate the solutions of SDRE. More specifically, the nonlinear optimal feedback control has been obtained by solving continuous algebraic Ricatti and Lyapunov equations. Sliding variables are designed such that reaching phase is eliminated and stability is guaranteed. The proposed control method equipped with high-order observer can guarantee more superior results than linear techniques such as linear quadratic regulator (LQR), conventional ISMC, and first-order sliding-mode control (SMC) method. Increasing the number of terms of the Taylor's series of the proposed control law provides better approximation, therefore the performance is improved. However, this increases the computational burden. The effectiveness of the control method is validated via simulations in MATLAB/Simulink under nominal parameters and model uncertainties.

Globally Optimal State-Feedback LQG Control for Large-Scale Systems With Communication Delays and Correlated Subsystem Process Noises

This paper studies the optimal decentralized state-feedback control of large-scale systems. The large-scale system is composed of subsystems and defined over a connected digraph. One step time is required for information to travel across an edge in the graph. Under the above-mentioned setup, when subsystem process noises are uncorrelated, the explicit optimal state-feedback controller can be designed by independence decomposition based on information hierarchy graph. However, this decomposition method fails when the subsystem process noises are correlated. In this paper, we propose a new decomposition method for system state and control input, and split the optimal state-feedback control problem with correlated process noises into two subproblems that can be solved separately. The solution to the first subproblem can be obtained by solving a linear matrix equation. The second subproblem is solved by algebraic Ricatti equation.

Feedback stabilization for one sided Lipschitz nonlinear systems in reciprocal state space: Synthesis and experimental validation

Abstract This paper proposes a design of a stabilization control for one-sided Lipschitz (OSL) nonlinear systems in new reciprocal state space (RSS) framework. The main objective is to extend the state derivative feedback stabilization methods for a class of nonlinear systems where the nonlinearity of derivatives state satisfies the OSL properties in RSS. The presented controller is composed of a state derivative feedback approach in order to ensure asymptotic stability in the sense of Lyapunov. The first approach deals with the synthesis of a basic controller by adopting a simple transformation of Linear Matrix Inequality (LMI) to standard algebraic Ricatti equation (ARE). The second is an extension to adaptive version with adjustment parameters. High performances are shown through real-time implementation with a hardware in the loop (HIL) mode using digital signal processing (DSP) device (DSpace DS 1104).

Minimum-Energy Distributed Consensus Control of Multiagent Systems: A Network Approximation Approach

This paper investigates the problem of minimizing the global energy cost for multiagent systems to achieve consensus over undirected network topologies. Conventional optimization problems so defined require all-to-all network topologies for interagent communications, which precludes the use of distributed control or otherwise demands that the network be complete. To circumvent this difficulty, we introduce a network approximation (NA) scheme in the optimization criterion, which tends to render the optimization problem to one seemingly over a complete graph network, thus removing the all-to-all communication requirement. With the NA cost, we show that a distributed optimal consensus algorithm always exists for any given connected network topology, which can be determined by solving a single-agent-level parametric algebraic Ricatti equation (PARE). We also investigate the performance of the optimal consensus algorithm, focusing on the minimal energy cost required to achieve consensus optimally, and the speed at which consensus is achieved. Furthermore, for certain more special yet worthy cases, such as single-integrator, double-integrator, and first-order unstable agents, we derive explicit expressions for the energy cost and the consensus speed. It can be seen from these results that the energy cost can be made arbitrarily small for single-integrator and double-integrator systems under the optimal distributed control. On the other hand, for the first-order unstable agents, the energy cost increases and consensus speed decreases monotonically with the value of the agent's real unstable pole.

Analytical and numerical methods for computing electron partial intensities in the case of multilayer systems

We present two novel methods for computing energy spectra and angular distributions of electrons emitted from multi-layer solids. They are based on the Ambartsumian–Chandrasekhar (AC) equations obtained by using the invariant imbedding method. The first method is analytical and relies on a linearization of AC equations and the use of the small-angle approximation. The corresponding solution is in good agreement with that computed by using the Oswald–Kasper–Gaukler (OKG) model, which is extended to the case of layers of finite thickness. The second method is based on the discrete ordinate formalism and relies on a transformation of the AC equations to the algebraic Ricatti and Lyapunov equations, which are solved by using the backward differential formula. Unlike the previous approach, this method can handle both linear and nonlinear equations.We analyze the applicability of the proposed methods to practical problems of computing REELS spectra. To demonstrate the efficiency of the proposed methods, several computational examples are considered. Obtained numerical and analytical solutions show good agreement with the experimental data and Monte-Carlo simulations. In addition, the impact of nonlinear terms in the Ambartsumian–Chandrasekhar equations is analyzed.

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.

Tracking control of leader-follower multi-agent systems subject to actuator saturation

This paper addresses the tracking problem of leader-follower multi-agent systems subject to actuator saturation. A leader node or command generator is considered, which generates the desired tracking trajectory. To track such a leader, a new family of scheduled low-and-high-gain feedback controllers is designed for each follower, provided that the linear dynamic mode is asymptotically null controllable with bounded controls, and such control laws rely on the asymptotic property of a class of parametric algebraic Ricatti equations. We show that if the associated undirected graph of the system is connected, with the proposed control law, all the followers can track the leader eventually. A simulation example is finally given to illustrate the performance of the proposed control scheme.

Nonlinear stabilization by dynamic parameter adaptation: Algebraic-Ricatti-Equation-based approach

A dynamic-gain parameterized controller is proposed to stabilize a class of nonlinear systems subject to norm-bounded uncertainties. The stabilizing controller is designed as a standard linear feedback with polynomial dynamic parameters. The expression of the dynamic parameters is defined from the solution of an Algebraic Ricatti Equation. The pendulum cart system with other examples are given as illustrative case studies to show the simplicity, the straightforwardness and the efficiency of the proposed design.

Consensus of linear multi-agent systems subject to actuator saturation

This paper is aimed at studying the consensus of linear multi-agent systems subject to actuator saturation. In order to solve the consensus problem, a new family of scheduled low-and-high-gain decentralized control laws are designed, provided that the dynamics of each agent is asymptotically null controllable with bounded controls, and such control laws rely on the asymptotic property of a class of parametric algebraic Ricatti equations. It is shown that the consensus of the systems with connected and fixed topology can be achieved semi-globally asymptotically via the local error low-and-high-gain feedback. An illustrative example with simulations shows that our method as well as control protocols is effective for the consensus of the linear multi-agent systems subject to actuator saturation.

DIRECTIONAL CHANGE ISSUES IN LQG CONTROL WITH ACTUATOR FAILURE

A discrete-time LQG control with actuator failure is considered with directional change taken into account, as a result of actuator saturation case included. The control problem is analyzed in terms of algebraic Ricatti equations. Results of numerous computer simulations of two-input two-output system are given to illustrate the performance of the reliable LQG controller.

Approximate dynamic programming solutions with a single network adaptive critic for a class of nonlinear systems

Approximate dynamic programming (ADP) formulation implemented with an adaptive critic (AC)-based neural network (NN) structure has evolved as a powerful technique for solving the Hamilton-Jacobi-Bellman (HJB) equations. As interest in ADP and the AC solutions are escalating with time, there is a dire need to consider possible enabling factors for their implementations. A typical AC structure consists of two interacting NNs, which is computationally expensive. In this paper, a new architecture, called the ‘cost-function-based single network adaptive critic (J-SNAC)’ is presented, which eliminates one of the networks in a typical AC structure. This approach is applicable to a wide class of nonlinear systems in engineering. In order to demonstrate the benefits and the control synthesis with the J-SNAC, two problems have been solved with the AC and the J-SNAC approaches. Results are presented, which show savings of about 50% of the computational costs by J-SNAC while having the same accuracy levels of the dual network structure in solving for optimal control. Furthermore, convergence of the J-SNAC iterations, which reduces to a least-squares problem, is discussed; for linear systems, the iterative process is shown to reduce to solving the familiar algebraic Ricatti equation.

A refined semilocal convergence analysis of an algorithm for solving the Ricatti equation

The semilocal convergence of a numerical algorithm for solving the algebraic Ricatti equation with multiparameter singularly perturbed systems is investigated here. We show that under weaker hypotheses and the some computational cost than in Mukaidani et al. (J. Math. Anal. Appl. 267:209–234, [2002]) finer estimates on the distances involved and a more precise information on the location of the solution can be obtained.

Non-linear systems control via fuzzy models: a multicontroller approach

This paper presents a Lyapunov-based switching controller design method for non-linear systems using Takagi-Sugeno fuzzy models. The basic idea of the proposed approach is to represent the fuzzy model as a set of uncertain linear systems. The controller is obtained by solving the corresponding set of Algebraic Ricatti Equations (AREs). A simulation example is given to illustrate the effectiveness of this approach.

ACTIVE CONTROL SIMULATION OF TORSIONAL VIBRATION FOR TURBINE-GENERATOR SHAFT SYSTEM BASED ON GLOBALLY OPTIMAL CONTROL ALGORITHMS OF QUADRATIC REGULATOR

Global optimal control algorithms of linear quadratic regulator is combined with torsional vibration theory of turbine- generator shaft system, and torsional vibration energy of shaft system and control energy are chosen as performance index of active control. The global optimal control theory with linear quadratic regulator is brought about to control the tortional vibration of the large turbine-generator shaft system. And space state equations on active control of torsional vibration of turbine-generator shaft system are constructed. A solution way on Algebraic Ricatti Equation is proposed to determine the feedback control input matrix. This method is to construct a Hamiltonian matrix and solve its eigenvalue. A simulation program on active control of torsional vibration for discrete system is compiled and simulation analysis on active control of torsional vibration is carried out in a shaft system of 200 MW turbine-generator to examine the active control effectiveness. The results of time domain analysis on active control of torsional vibration show that employing the global optimal control theory of linear quadratic regulator can effectively decrease torsional vibration and control torque input.

A note on the necessary conditions for the algebraic Riccati equation

We derive some useful and easily computable necessary conditions for the existence of a positive semi-definite solution to the algebraic Ricatti equation (ARE). A motivating example is given to highlight the usefulness of the conditions for controller and observer designs for nonlinear systems. Further, an upper bound on the trace of the solution to the ARE is also derived.

An algorithm to solve the discrete HJI equation arising in the L2 gain optimization problem

A synthesis of the discrete non-linear H control law boils down to the solution of a set of algebraic and partial differential equations known as the discrete Hamilton-Jacobi-Isaacs (DHJI) equation, an extension of the discrete algebraic Ricatti equation arising in the linear discrete H control problem. Due to its non-linear nature, it is rarely possible to obtain the closed form solution for DHJI equation. This paper proposes an approximation approach to solving the DHJI equation in terms of the Taylor series. It is shown that the coefficients of the Taylor series solution for DHJI equation are governed by one discrete algebraic Ricatti equation and a sequence of linear algebraic equations, respectively. This result lends itself to a systematic algorithm to approximately synthesize a discrete non-linear H control law.

Transformation of Nonlinear Systems into State Affine Control Systems and Observer Synthesis

This paper addresses the synthesis of observers for nonlinear systems and their equivalence to state affine systems (i.e. time-varying linear systems). Necessary and sufficient conditions are given and then, whether a solution exists, one gets the observer synthesis. These constructive conditions are based on the analysis of thesystem input/output (I/O) differential equation. The transformed system is again transformed to solved an algebraic Ricatti 's equation instead of a time varying one.

Suboptimal feedback control for a class of nonlinear systems using spline interpolation

We consider a class of nonlinear quadratic regulator problems where the system dynamics are affine in the control. It has been shown recently that an optimal feedback control law for this class of problems can be given in terms of the solution of a state dependent algebraic Ricatti equation (ARE) at each instance of time. However, in most practical problems it is not possible to find an analytic solution to the ARE and hence numerical schemes to calculate suboptimal controls are required. In this paper, we consider one such scheme based on cubic basis spline interpolation. It is shown that if the chosen partitioning of the state space is sufficiently small, the resulting suboptimal controller leads to a stable closed loop system.