Showing papers on "Riccati equation published in 1975"
01 Sep 1975
TL;DR: In this paper, a cost function which incorporates system input, output and set-point variations is selected, and a control law for a known system is derived, and the control input is chosen to make the prediction zero.
Abstract: A strategy for the design of self-tuning controllers of systems with constant but unknown parameters is presented. A cost function which incorporates system input, output and set-point variations is selected, and a control law for a known system is derived. This control law is shown to comprise a least-squares predictor of a function related to the cost function, and the control input is chosen to make the prediction zero. The parameters of the control law for the unknown system are estimated using a recursive-least-squares algorithm, and the optimal parameters are shown to be a fixed point of the algorithm. Whilst retaining their computational simplicity, the proposed method has several advantages over self-tuning-regulator strategies which attempt to minimise the output variance alone: weighting of control is allowed for; set-point variation may be optimally followed; there is no requirement to choose a system-related parameter to ensure convergence; and, for stable but nonminimum phase systems, there is no need to employ time-consuming methods, such as the solution of a Riccati equation. Several simulated examples are used to demonstrate the potential of the method.
770 citations
609 citations
TL;DR: The quasi-normal modes of a black hole represent solutions of the relevant perturbation equations which satisfy the boundary conditions appropriate for purely outgoing (gravitational) waves at infinity and purely ingoing waves at the horizon as discussed by the authors.
Abstract: The quasi-normal modes of a black hole represent solutions of the relevant perturbation equations which satisfy the boundary conditions appropriate for purely outgoing (gravitational) waves at infinity and purely ingoing waves at the horizon. For the Schwarzschild black hole the problem reduces to one of finding such solutions for a one-dimensional wave equation (Zerilli's equation) for a potential which is positive everywhere and is of short-range. The notion of quasi-normal modes of such one-dimensional potential barriers is examined with two illustrative examples; and numerical solutions for Zerilli's potential are obtained by integrating the associated Riccati equation.
508 citations
277 citations
TL;DR: In this article, a simple transformation, originally introduced for singularly perturbed systems, is now applicable to a larger class of time-invariant systems and applied to a large class of systems.
Abstract: A simple transformation, originally introduced for singularly perturbed systems, is now applicable to a larger class of time-invariant systems.
209 citations
Book•
01 Jan 1975
206 citations
TL;DR: In this article, a suboptimal solution to the nonlinear quadratic regulator and tracking problem with infinite final time is investigated, and it is shown that with certain restrictions the sub-optimal control law exists and is a continuous function of state and time.
156 citations
TL;DR: In this article, an exploratory study is made to investigate the feasibility of applying such a theory to control the vibration of civil engineering structures under random loadings, where it is assumed that random excitations to structures, such as wind loads and earthquakes, can be modeled by passing either a stationary Gaussian white noise or a nonstationary Gaussian shot noise through a filter.
Abstract: Modern control theory has been successfully applied to control the motions of aerospace vehicles. An exploratory study is made herein to investigate the feasibility of applying such a theory to control the vibration of civil engineering structures under random loadings. It is assumed that random excitations to structures, such as wind loads and earthquakes, can be modeled by passing either a stationary Gaussian white noise or a nonstationary Gaussian shot noise through a filter. The performance index to be minimized consists of the covariances of both the structural responses and the control forces. Under these conditions, the optimal control law is a linear feedback control. The optimal control forces are obtained by solving a matrix Riccati equation. Applications of the optimal control to a multi-degree-of-freedom structure, under stationary wind loads and nonstationary earthquakes, are demonstrated. It is shown that significant reduction in covariances of the structural responses can be achieved by the use of an active control system.
149 citations
TL;DR: In this article, the authors studied the infinite-time quadratic cost control problem for a general class of linear autonomous hereditary differential systems using an approach which clarified the system-theoretic relationship between stabilizability, stability and existence of a solution of an associated operator equation of Riccati type.
Abstract: This paper studies the infinite-time quadratic cost control problem for a general class of linear autonomous hereditary differential systems. It uses an approach which clarifies the system-theoretic relationship between stabilizability, stability and existence of a solution of an associated operator equation of Riccati type. For this purpose the stability problem is studied and an operator equation of the Lyapunov type is derived. In both cases we obtain equations which characterize the kernels of the Lyapunov and the Riccati equations.
136 citations
TL;DR: In this paper, the existence and uniqueness of the inner product algebraic Riccati equation of optimal control in Hilbert space is proved under certain assumptions and some related problems such as stabilizability and exact controllability of control systems are investigated.
Abstract: Under certain assumptions the existence and uniqueness of the solution of the inner product algebraic Riccati equation of optimal control in Hilbert space are proved. Some related problems such as stabilizability and exact controllability of control systems are investigated also.
119 citations
TL;DR: In this paper, the authors extended the existence theory of the stabilizing solution of the discrete algebraic Riccati equation and provided necessary and sufficient conditions for the existence of such a solution, free of any a priori conditions on the problem data.
Abstract: This short paper extends the existence theory of the stabilizing solution of the discrete algebraic Riccati equation. The main result provides necessary and sufficient conditions for the existence of such a solution, free of any a priori conditions on the problem data. For the special case of the optimal regulator problem, standard results are recovered as special cases.
TL;DR: In this article, the periodic solution of matrix Riccati differential equations with periodic coefficients was studied and it was shown that the existence of a periodic solution is equivalent to detectability and stabilizability of certain coefficient pairs.
Abstract: This paper discusses the periodic solution of matrix Riccati differential equations with periodic coefficients. Such equations arise in linear filtering and control and in many other applications. The principal result: the existence of a periodic solution is equivalent to detectability and stabilizability of certain coefficient pairs. This result generalizes the Kalman–Wonham–Kucera theorem for algebraic Riccati equations. Among the numerous preliminaries is a discussion, apparently new, of detectability for linear periodic control systems. Another important result, for a linear matrix differential equation, is the equivalence of a bounded solution, an exponentially stable solution and a periodic solution. Finally, the periodic solution is shown to be an equilibrium solution in the sense of Kalman.
TL;DR: In this paper, the existence and uniqueness of the Riccati operator difference equation and the asymptotic behavior of the solution of such an equation are investigated under certain assumptions.
Abstract: A general system described by a linear difference equation in a Hilbert space is considered. Three types of disturbances, control-dependent noise, state-dependent noise and purely additive noise, are taken into account. The cost function is assumed to be quadratic. The existence of an optimal stationary strategy and the uniqueness of the stationary measure related to this strategy are proved.Special attention is paid to the related Riccati operator difference equation and the asymptotic behavior of the solution of such an equation is investigated. Under certain assumptions, the existence and uniqueness of the solution of the algebraic Riccati equation are proved, too.
TL;DR: In this article, a general existence theorem for the equilibrium points of the Riccati: equation is proven and their structure is given with respect to the partial ordering induced by the positive semi-definite matrices.
Abstract: : A study of the matrix Riccati equation is presented without assuming that Q is positive semi-definite. For the autonomous case, a general existence theorem for the equilibrium points of the Riccati: equation is proven and their structure is given with respect to the partial ordering induced by the positive semi-definite matrices. Further, a method for finding all equilibria is demonstrated. Necessary and sufficient conditions are given for a global existence of the solution of the Riccati equation, as a function of the initial condition. In the time dependent case, the domain of global existence is shown to contain a certain cone.
01 Dec 1975
TL;DR: Backward and forward-time differentiations are introduced that readily provide the generalized Chandrasekhar algorithms as well as several interesting interpretations of these results.
Abstract: The Riccati equation (RE) plays a fundamental role in optimal control theory, linear estimation, radiative transfer, neutron transport theory, etc. Its effective, numerical solution constitutes the integral prerequisite to the solution of important problems in the above and related fields. A computationally advantageous approach to the solution of matrix Re's is the so-called x-y or Chandrasekhar algorithm through which the matrix RE is replaced by two coupled differential equations of lesser dimensionality. These previous Chandrasekhar algorithms were, however, restricted to the case of time-invariant models. In this short paper, generalized x-y or Chandrasekhar algorithms are presented that are applicable to time-varying models as well as time-invariant ones. Backward and forward time differentiations are introduced that readily yield the generalized Chandrasekhar algorithms as well as provide several interesting interpretations of these results. Furthermore, the possible computational advantages, as well as the theoretical significance of the generalized Chandrasekhar algorithms are explored.
TL;DR: It is proved that every state that is sufficiently small in a sense made precise in the paper can be reached from the origin in a timeT depending on the coefficients of the equation.
Abstract: We obtain results on local controllability (near an equilibrium point) for a nonlinear wave equation, by application of an infinite-dimensional analogue of the Lee-Markus method of linearization. Controllability of the linearized equation is studied by application of results of Russell, and local controllability of the nonlinear equation follows from the inverse function theorem. We prove that every state that is sufficiently small in a sense made precise in the paper can be reached from the origin in a timeT depending on the coefficients of the equation.
TL;DR: “Instantaneous” linear and square laws that are “local” conditions for winning are shown to apply for Lanchester-type formulations with time-varying system effectiveness, and qualitative insight is provided as to the “direction” in which combat is moving.
Abstract: This paper studies the Riccati equation satisfied by the force ratio of two homogeneous forces in deterministic Lanchester combat. This study provides qualitative insight as to the “direction” in which combat is moving (in the sense that the force ratio is changing to the advantage of one of the combatants). “Instantaneous” linear and square laws that are “local” conditions for winning are shown to apply for Lanchester-type formulations with time-varying system effectiveness. The value of considering the force-ratio equation lies in one's ability to determine the outcome of battle without explicitly solving the force-level equations, an important feature for variable-coefficient formulations. Moreover, in some cases observation of the initial time-behavior of the force ratio in a battle allows one to predict the outcome, sometimes even without explicit knowledge of attrition-rate coefficients. By additionally considering the instantaneous loss ratio, one gains insight into the consequences of concentratio...
TL;DR: It is proved the existence of a non-constant periodic solution under the conditions birth rate b > π/2 and τ- 1 small enough.
Abstract: We consider an integro-differential equation for the densityn of a single species population where the birth rate is constant and the death rate depends on the values ofn in an interval of length τ - 1 > 0. We prove the existence of a non-constant periodic solution under the conditions birth rate b > π/2 and τ- 1 small enough. The basic idea of proof (due to R. D. Nussbaum) is to employ a theorem about non-ejective fixed points for a translation operator associated with the solutions of the equation.
01 Jan 1975
TL;DR: In this article, the effects of stiffness on production codes for solving non-stiff ODEs are investigated, and a practical test for stiffness in variable order Adams codes is developed.
Abstract: The effects of stiffness are investigated for production codes for solving non-stiff ordinary differential equations. First, a practical view of stiffness as related to methods for non-stiff problems is described. Second, the interaction of local error estimators, automatic step size adjustment, and stiffness is studied and shown normally to prevent instability. Third, a practical test for stiffness in variable order Adams codes is developed. Numerical results for working codes are presented for each topic.
TL;DR: In this paper, robust and fast computational algorithms for the solution of discrete Riccati equations (RE) are presented using the "partitioning" approach to estimation and control, which results through partitioning the total computation interval into subintervals and solving for the RE in each subinterval with zero initial conditions for each sub-interval.
Abstract: 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.
TL;DR: In this article, a generalized version of the X-Y functions of radiative transfer is given, and it is seen that, under commonly occurring conditions, the new equations may be easier to numerically resolve than the original Riccati equation.
Abstract: The operator Riccati equation associated with a distributed parameter quadratic cost-linear dynamics control process is considered.Making use of ideas from transport theory, a derivation of a generalized version of the X-Y functions of radiative transfer is given, and it is seen that, under commonly occurring conditions, the new equations may be easier to numerically resolve than the original Riccati equation. In particular, the new equations express directly the optimal gain function. The analytic results are illustrated by a numerical example of heat regulation on a rod.
TL;DR: The problem of assigning traffic in a multiserver system, where customer-server relationships are determined beforehand, can be regarded as a problem in system design, and designs are identified which lead to optimization of several alternative criteria.
Abstract: The problem of assigning traffic in a multiserver system, where customer-server relationships are determined beforehand, can be regarded as a problem in system design. This is done, and designs are identified which lead to optimization of several alternative criteria. The notion of most efficient set of servers is presented ; some ramifications, into different forms of scheduling, are discussed.
01 Jun 1975
TL;DR: In this paper, the Riccati equation arises in a natural family of equations evolving forwards as well as backwards in time, which allows an interesting derivation of the fast Chandrasekhar-type equations for linear least-squares filtering of processes generated by a time-invariant, finite-dimensional linear system driven by white noise.
Abstract: : In scattering theory the Riccati equation arises in a natural family of equations evolving forwards as well as backwards in time. The authors show how this framework allows an interesting derivation of the fast Chandrasekhar-type equations for linear least-squares filtering of processes generated by a time-invariant, finite-dimensional linear system driven by white noise. The processes are not required to be stationary. The same ideas can be used to obtain Levinson- and Cholesky-type differential equations for the impulse responses of the whitening filter and the innovations representation of such processes. The scattering framework brings out clearly both the significance of the time-invariance of the parameters of the underlying finite-dimensional system and of the associated family of nonstationary processes. For stationary processes, it also becomes clear that the assumption of finite-dimensionality is unnecessary, but the proper extension of the nonstationary class of processes raises some interesting questions.
TL;DR: A review and comparative study of the various optimisation methods applied to electrical power systems and it is shown that the suboptimal control using feedback of a selected number of variables derived by Liapunov method can be equally effective.
Abstract: A review and comparative study of the various optimisation methods applied to electrical power systems is made in this paper. The linear optimal control derived by solving the matrix Riccati equation requires feedback of all the state variables. It is shown that the suboptimal control using feedback of a selected number of variables derived by Liapunov method can be equally effective. Under certain conditions optimal control derived from lineamised system equations can give improved system perfommance following large disturbances. However to achieve maximum improvement it is necessary to retain all the system nonlinearities in the optimisation procedure. A method of optimising a system for a nonlinear mode of operation retaining the advantages of closed loop control is described. Finally, the feasibility of practical implementation is investigated by incorporating the various optimal controls derived in a laboratory micromachine system.
TL;DR: In this paper, the authors present a new proof that aq-step backward difference scheme for the approximate solution of a first order ordinary differential equation is stable in the sense of Dahlquist iff 1≦q≦6.
Abstract: We present a new proof that aq-step backward difference scheme for the approximate solution of a first order ordinary differential equation is stable in the sense of Dahlquist iff 1≦q≦6.
TL;DR: The theory of periodic generators of the Riccati equation is presented in this article, as well as a method for their determination. But this method is not suitable for the case where the generator is unknown.
Abstract: The theory of periodic generators of the Riccati equation is presented, as well as a method for their determination
TL;DR: In this paper, the authors apply the theory to the linear quadratic cost control problem for the afBne linear hereditary differential case and apply it to the semigroup formulation of linear hereditary equations introduced by Delfour and Mitter.
Abstract: The infinite-dimensional versions of the linear quadratic cost control problem and of the linear filtering problem lead to an infinite-dimensional Riccati equation with unbounded operators. Existence and uniqueness theorems for mild solutions of these were established in The infinite dimensional Riccati equations, Ruth F. Curtain and A. J. Pritchard (to appear in J. Math. Anal. Appl.) using a semigroup and evolution operator approach. Although this formulation was very general, covering a large class of parabolic partial differential control systems, it does not cover the semigroup formulation of linear hereditary differential equations introduced by Delfour and Mitter. This paper remedies this and applies the theory to the linear quadratic cost control problem for the afBne linear hereditary differential case.
TL;DR: In this article, the critical length (escape time) associated with a solution of a matrix Riccati equation is shown to be bounded in terms of a known value of the solution at any point.
01 Jan 1975
TL;DR: In this article, it was shown that if f is an entire function and satisfies a certain differential equation, then it is shown that f is of bounded index, which extends a theorem of S. M. Shah.
Abstract: If f is an entire function and satisfies a certain differential equation, then it is shown that f is of bounded index. This extends a theorem of S. M. Shah.