Riccati equation solution for controllers with continuous delays

We reduce the solution of a Riccati equation for infinite-time linear quadratic controllers with continuous delays and n state variables to the problem of finding scalar parameter values for an integral kernel whose form is completely specified. To simplify the exposition, the reduction is described only for a special case involving 2-dimensional state variables, but the method is entirely general. An abstract formulation of Vinter and Kwong is used throughout.     

A Riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems

This paper presents a procedure for designing a full state observer and feedback control law which will stabilize a given uncertain linear system. The uncertain linear systems under consideration are described by state equations which depend on uncertain parameters. These uncertain parameters may be time varying. Their values, however, are constrained to lie within known compact bounding sets. The design procedure involves solving two algebraic Riccati equations. A feature of the design procedure presented is the fact that it reduces to the standard LQG design procedure if the system contains no uncertain parameters.     

The algebraic Riccati equation — A polynomial approach

Polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic Riccati equation.     

An efficient approach for solving the Riccati equation with fractional orders

The present study introduces a novel and simple analytical method for the solution of fractional order Riccati differential equation. In this approach, the solution considered as a Taylor series expansion converges rapidly to the nonlinear problem. New homotopy perturbation method (NHPM) depends only on two components of the homotopy series. The method is illustrated by applications and the results obtained are compared with those of the exact solution. Moreover, comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient.     

Memoryless stabilization of uncertain dynamic delay systems:Riccati equation approach

The authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems. The uncertainties are time varying and within a given compact set. This method is an extension of the Riccati equation approach proposed by I.R. Petersen and C.V. Hollot (1986). The extension is straightforward. Also the uncertainties do not need to satisfy the matching conditions     

Designing stabilizing controllers for uncertain systems using theRiccati equation approach

A useful technique for determining a linear feedback control law stabilizes an uncertain system is the Riccati-equation approach of I.R. Petersen and C.V. Hollot (1986). They consider systems with time-varying uncertainty in the system matrix and obtain the constant feedback gains for the linear stabilizing controller in terms of the solutions of a Riccati equation. The technique is extended to include problems with time-varying uncertainty in the input connection matrix. Several examples are included to demonstrate the efficacy of this result     

On the discrete Riccati equation

Several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix Riccati equation. This note adds an alternative one to them.     

A square root of a matrix approach to obtain the solution to a steady-state matrix Riccati equation

By utilizing the square root of a matrix approach, a method of generating the solution to a steady-state matrix Riccati type equation (under certain restrictions) is presented. This approach not only yields a closed form expression for the Riccati solution, but also converts the original Riccati equation into other equations which may have numerical or computational advantages. An example is worked out for a second-order case.     

On the design of multivariable PID controllers via LMI approach

In this paper, we study the design problem of multivariable PID controllers which guarantee the stability of the closed loop systems, H₂ or H_∞ performance specifications, or maximum output control requirement, respectively. Algorithms based on iterative linear matrix inequality technique are developed to find the feedback gains of PID controllers corresponding to the above mentioned four cases. A numerical example on the design of PID controllers for aircraft is provided to illustrate the effectiveness of the proposed method.     

A new stochastic approach for solution of Riccati differential equation of fractional order

In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.     

Quadratic guaranteed cost control for uncertain dissipative models: A Riccati equation approach

The problem of H₂ guaranteed cost control and dynamic output-feedback for linear uncertain systems with dissipative uncertainty is addressed. The problem of robust H₂ synthesis has been open for the last two decades. In this paper, a problem of H₂ quadratic guaranteed cost control is defined for uncertain systems affected by LTI quadratic dissipative model uncertainty. A necessary and sufficient condition of quadratic stabilizability via output-feedback is derived in terms of two coupled parameter-dependent Riccati equations. Then, a method is given to design controllers which minimize an upper bound for the worst-case H₂ norm of the uncertain system. It therefore assesses a guaranteed level of robust performance where in literature, only nominal performance is ensured in most cases. A reliable numerical iterative procedure based on Riccati solvers and one-dimensional convex parameter search is provided. With this uncertainty modelling and the developed numerical procedure, we hope to reduce the usual conservatism of quadratic designs.     

Riccati equation in projecting solenoidal fields

Unexpected relationship between the Riccati equation of the Layer Stripping Method and the projecting solenoidal vector fields is found out. A transform which diagonalizes a family of projectors on increasing subspaces of solenoidal fields is constructed. Due to the Ricatti equation the transform turns out to be isometrical.     

Algebraic Riccati equation and symplectic algebra

Questions of existence, uniqueness and the parametric dependence of solutions of the algebraic Riccati equation are considered. The different criteria for the solubility of this equation are obtained with the help of symplectic algebra.     

Dead-beat control and the Riccati equation

Connections between dead-beat control strategies and optimal control policies for linear, time-invariant, discrete-time systems are established. The performance index of the system is quadratic and only the terminal state of the system is penalized. An explicit solution to the singular Riccati equation, associated with this optimization problem, is given. Properties of the time-variable gain matrices, generating the optimal cantrol policy, are presented. In particular, necessary and sufficient conditions for each of these gain matrices to be a time-invariant deadbeat controller are given.     

Algorithms of constructing a stabilizing solution of the algebraic Riccati equation based on its linear reduction in the problems of analytical construction of controllers

In the problem of the stabilizing solution of the algebraic Riccati equation, the resolvent Θ(s) = (s I _2n ? H)^?1 of the Hamilton 2n × 2n-matrix H of the algebraic Riccati equation allows us to reduce the problem to a linear matrix equation. In [1], the constructions necessary for this and the theorem of existence and representation of the stabilized solutions to an algebraic Riccati equation was proposed. In this paper, the methods of constructing the resolvent and the linear reduction matrix defined by it necessary for the application of the theorem, and in addition, the algorithms of constructing stabilizing solution of the algebraic Riccati equation are proposed.     

Curvature properties of the algebraic Riccati equation

We prove that the solution to the algebraic Ricatti equation (ARE) is concave with respect to a nonnegative-definite symmetric state weighting matrix Q when the input weighting matrix R = R^T > 0. We also prove that the solution to the ARE is concave with respect to a positive-definite diagonal input weighting matrix R when Q = Q^T ≥ 0.     

Discrete Riccati equation solutions: Partitioned algorithms

Using the "partitioning" approach to estimation and control, robust and fast computational algorithms for the solution of discrete Riccati equations (RE) are presented. The algorithms have a decomposed or partitioned structure that results through partitioning the total computation interval into subintervals and solving for the RE in each subinterval with zero initial conditions for each subinterval Thus, effectively, the RE solution over the whole interval has been decomposed into a set of elemental piece-wise solutions which are both simple as well as completely decoupled from each other and as such computable in either a parallel or serial processing mode. Further, the overall solution is given in terms of a simple recursive operation on the elemental solutions. The partitioned algorithms are theoretically interesting as well as computationally attractive.     

On Newton's method for Riccati equation solution

The purpose of this note is to point out that the assumptions of two theorems of Kleinman concerning Newton's method for the Riccati equation can be weakened.     

Nonexistence of solutions to the Riccati equation

The Riccati equation does not necessarily give a positive definite solution when the plant has finite escape time in the time interval considered.     

Global aspects of the matrix Riccati equation

Matrix Riccati equations are interpreted as differential equations on Grassman manifolds. Necessary conditions for the Riccati equation to be a Morse-Smale system are given in the autonomous and periodic cases. Under this condition, the equation is structurally stable and has a unique asymptotically stable equilibrium point or periodic solution.