Showing papers on "Riccati equation published in 1987"
TL;DR: In this paper, the authors present an algorithm for the stabilization of a class of uncertain linear systems, which is described by state equations which depend on time-varying unknown-but-bounded uncertain parameters.
1,483 citations
TL;DR: This paper solves a general finite-horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor, codescriptor, and control vectors, and calculates the optimal feedback gain relating the control and descriptor variable.
Abstract: In this paper we investigate the linear-quadratic optimal regulator problem for the continuous-time descriptor system E\dot{x} = Ax + Bu where E is, in general, a singular matrix. We solve first a general finite-horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor, codescriptor, and control vectors. From this trajectory the optimal feedback gain relating the control and descriptor variable can be computed. By transforming to a coordinate system which can be computed by performing a singular value decomposition of E we derive several Riccati differential equations, all of which have the same solution; this solution gives the optimal cost. The steady-state optimal feedback gain can be computed by solving an eigenvalue-eigenvector problem formulated from the untransformed system parameters. In general, there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of E .
372 citations
TL;DR: In this paper, a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system is presented. But this method requires the solution of a certain algebraic Riccati equation.
Abstract: This note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system. The problem under consideration involves attenuating the disturbances to a prespecified level. The construction of the state feedback control law requires the solution of a certain algebraic Riccati equation. By applying the procedure with successively smaller values of the prespecifled disturbance level, the result also provides an approach to the H^{\infty} optimization problem for the state feedback case.
366 citations
TL;DR: In this article, the problem of assigning all poles of a closed-loop system in a specified disk by state feedback is considered for both continuous and discrete systems, and a state feedback control law is determined by using a discrete Riccati equation.
Abstract: The problem of assigning all poles of a closed-loop system in a specified disk by state feedback is considered for both continuous and discrete systems. A state feedback control law is determined by using a discrete Riccati equation. This kind of pole assignment problem is named D -pole assignment, and its relation to the optimal control problem and its robustness properties are discussed. The gain and phase margins for all closed-loop poles to stay inside the specified disk D are determined for the proposed control.
281 citations
TL;DR: This paper solves a somewhat more general finite-horizon problem by applying Hamiltonian minimization to derive the optimal trajectory of the vector consisting of the concatenated descriptor, codescriptor, and control vectors and derives a Riccati differential equation whose solution gives the optimal cost.
270 citations
TL;DR: In this article, the matrix-sign-function algorithm for algebraic Riccati equations is improved by a simple reorganization that changes nonsymmetric matrix inversions into symmetric matrix inverse inversions.
212 citations
TL;DR: In this article, a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators is established, which is similar to our framework.
Abstract: This paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators.
202 citations
TL;DR: In this article, the sufficiency tests are applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems, and the modified Riccati equation approach of Petersen and Hollot is generalized in the static case and extended to dynamic compensation.
Abstract: Three parallel gaps in robust feedback control theory are examined: sufficiency versus necessity, deterministic versus stochastic uncertainty modeling, and stability versus performance. Deterministic and stochastic output-feedback control problems are considered with both static and dynamic controllers. The static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time-varying parameter variations, while the static and dynamic stochastic optimal control problems feature state-, control-, and measurement-dependent white noise. General sufficiency conditions for the deterministic problems are obtained using Lyapunov's direct method, while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion. The sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems. As an additional application of the deterministic result, the modified Riccati equation approach of Petersen and Hollot is generalized in the static case and extended to dynamic compensation.
130 citations
TL;DR: Three methods for refining estimates of invariant subspaces are compared by changing variables that they all solve the same equation, the Riccati equation, and this shows a hybrid algorithm combining advantages of all three is suggested.
Abstract: We compare three methods for refining estimates of invariant subspaces, due to Chatelin, Dongarra/Moler/Wilkinson, and Stewart. Even though these methods all apparently solve different equations, we show by changing variables that they all solve the same equation, the Riccati equation. The benefit of this point of view is threefold. First, the same convergence theory applies to all three methods, yielding a single criterion under which the last two methods converge linearly, and a slightly stronger criterion under which the first algorithm converges quadratically. Second, it suggest a hybrid algorithm combining advantages of all three. Third, it leads to algorithms (and convergence criteria) for the generalized eigenvalue problem. These techniques are compared to techniques used in the control systems community.
87 citations
TL;DR: In this paper, the conditions suffisante for que toutes les solutions soient oscillatoires are discussed. And they utilise les techniques de Riccati et les principes variationnels.
Abstract: On etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires. On utilise les techniques de Riccati et les principes variationnels
82 citations
TL;DR: Given a transfer matrix described by a minimal state-space triple, a method is given for computing state- space realizations for the numerator and denominator of a normalized, stable, right coprime factorization for the transfer matrix using an algebraic Riccati equation.
Abstract: Given a transfer matrix described by a minimal state-space triple, a method is given for computing state-space realizations for the numerator and denominator of a normalized, stable, right coprime factorization for the transfer matrix. The method involves the solution of an algebraic Riccati equation. It allows the use of existing computational state-space algorithms in finding normalized stable right coprime factorizations, and avoids explicit calculations of spectral factors.
TL;DR: In this article, the authors considered the linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces, where the optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE).
Abstract: The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.
TL;DR: A new spline approximation scheme for retarded functional differential equations that preserves the product space structure of retarded systems and approximates the adjoint semigroup in a strong sense and guarantees the convergence of the solution operators for the differential Riccati equation.
Abstract: The purpose of this paper is to introduce a new spline approximation scheme for retarded functional differential equations. The special feature of this approximation scheme is that it preserves the product space structure of retarded systems and approximates the adjoint semigroup in a strong sense. These facts guarantee the convergence of the solution operators for the differential Riccati equation in a strong sense. Numerical findings indicate a significant improvement in the convergence behaviour over both the averaging and the previous spline approximation scheme.
TL;DR: In this paper, the authors considered the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain Ω, with control function acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes the L 2 (0, ∞; L 2(Ω))-norm of the solutiony and the L 1.2(0, ε) norm of the controlu.
Abstract: This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain Ω, with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL
2(0, ∞; L2(Ω))-norm of the solutiony and theL
2(0, ∞; L2(Γ))-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu
0, y0. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.
TL;DR: In this article, a new proof for the inequality tr (XY) \leq \parallel X ∈ {2} \cdot tr Y under the condition that X may be any square matrix.
Abstract: A new proof is presented for the inequality, tr (XY) \leq \parallel X \parallel_{2} \cdot tr Y . This argument is valid under the condition that Y be real symmetric nonnegative definite; X may be any square matrix.
TL;DR: A program that replaces a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use of lower-order lower orders was introduced by Riccati as mentioned in this paper.
Abstract: A program that dates back to Riccati [1724] and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use ...
TL;DR: In this article, the existence and uniqueness result on periodic solutions of an infinite dimensional Riccati equation was proved. But this result is not applicable to the periodic solution of the Riemann equation.
Abstract: We prove an existence and uniqueness result on periodic solutions of an infinite dimensional Riccati equation.
TL;DR: In this paper, sufficient conditions are derived which assure the existence of state feedback control laws such that the closed-loop system is locally and globally stable in probability, based on the solution of the stochastic algebraic Riccati equation.
Abstract: This paper deals with non-linear stochastic systems with state-dependent noise. Based on the solution of the stochastic algebraic Riccati equation, sufficient conditions are derived which assure the existence of state feedback control laws such that the closed-loop system is locally and globally stable in probability.
TL;DR: The method is close to the symplectic method for finding all the eigenvalues of a Hamiltonian matrix and is based on a (Γ, ΓG)-orthogonal transformation, which preserves structure and has desirable numerical properties.
TL;DR: A two-time scale discrete control system is considered and a composite, closed-loop suboptimal control is created from the sum of the slow and fast feedback optimal controls.
Abstract: A two-time scale discrete control system is considered. The closed-loop optimal linear quadratic (LQ) regulator for the system requires the solution of a full-order algebraic matrix Riccati equation. Alternatively, the original system is decomposed into reduced-order slow and fast subsystems. The closed-loop optimal control of the subsystems requires the solution of two algebraic matrix Riccati equations of order lower than that required for the full-order system. A composite, closed-loop suboptimal control is created from the sum of the slow and fast feedback optimal controls. Numerical results obtained for an aircraft model show a very close agreement between the exact (optimal) solutions and computationally simpler composite (suboptimal) solutions. The main advantage of the method is the considerable reduction in the overall computational requirements for the closed-loop optimal control of digital flight systems.
TL;DR: In this paper, an algebraic Riccati equation is studied for optimal control of deterministic and stochastic parabolic systems with boundary control, where the control function can act on the boundary through Dirichlet or Neumann conditions.
Abstract: An algebraic Riccati equation is studied, with application to the optimal control of deterministic and stochastic parabolic systems with boundary control. The control function can act on the boundary through Dirichlet or Neumann conditions.
TL;DR: The complete list of non-symmetric eight-vertex constant solutions to the Yang-Baxter equation is obtained and new solutions dependent on more than one variable are presented in this article.
Abstract: The complete list of non-symmetric eight-vertex constant solutions to the Yang-Baxter equation is obtained and new solutions dependent on more than one variable are presented. The corresponding spin Hamiltonians are derived by Sutherland's method (1970).
10 Jun 1987
TL;DR: In this article, the Riccati equation approach is extended to include problems with time-varying uncertainty in the input connection matrix, and the results for linear feedback control law with constant feedback gain are presented.
Abstract: A useful technique for determining a linear feedback control law which stabilizes an uncertain system is the Riccati equation approach of Petersen and Hollot [1,2]. 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 solution of a Riccati equation. Here we extend this technique to include problems with time-varying uncertainty in the input connection matrix. Several examples are included to demonstrate the efficacy of this result.
TL;DR: In this article, a control scheme for effective stabilization of discrete-time, nonlinear stochastic systems where the nonlinearity involves a zero-mean, independent random sequence is presented.
Abstract: A control scheme is presented for effective stabilization of discrete-time, nonlinear stochastic systems where the nonlinearity involves a zero-mean, independent random sequence. The control is of a constant feedback type and makes use of finite time solutions of a Riccati-like matrix difference equation. Stability results are both in terms of mean square and almost sure stochastic stability. Moreover, a matrix inequality is given to check the existence of weighting matrices which would result in a stable regulator problem.
TL;DR: An a posteriori estimate of the relative error in the solution of the algebraic matrix Riccati equation, found by the Schur approach, is derived and it is shown that in some cases theSchur approach is numerically unstable.
TL;DR: In this article, the trace of the solution to the discrete algebraic matrix Riccati equation has been shown to have a bound on the number of elements in the trace, which is the smallest bound known.
Abstract: 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.
TL;DR: In this article, a generalized frequency-shaped LQ theory is developed for plants with matrix fraction descriptions, and a spectral factorization based construction procedure is given which leads to stable output feedback controllers that are optimal in an LQ sense.
TL;DR: In this paper, the authors considered the case when G = SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k-dimensional vector space invariant (1≤k≤n).
Abstract: A system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group–subgroup pair G⊇G0. We consider the case when G=SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k‐dimensional vector space invariant (1≤k≤n). The nonlinear ordinary differential equations (ODE’s) in this case are rectangular matrix Riccati equations for a matrix W(t)∈Cn×k. The special case n=rk (n,r,k∈N) is considered and a superposition formula is obtained, expressing the general solution in terms of r+3 particular solutions for r≥2, k≥2. For r=1 (square matrix Riccati equations) five solutions are needed, for r=n (projective Riccati equations) the required number is n+2.
TL;DR: In this paper, the asymptotic expansion of the Korteweg-de Vries-Burgers equation is presented, and the expansion is shown to be linear in the number of Ks.
Abstract: The asymptotic expansion of the Korteweg-de Vries-Burgers equation is presented in this paper.
TL;DR: The introduction of stochastic notions of stabilizability and detectability gives a direct characterization of the asymptotic behaviour of the optimal system.