Stabilizability and positiveness of solutions of the jump linear quadratic problem and the coupled algebraic Riccati equation

This note addresses the jump linear quadratic problem of Markov jump linear systems and the associated algebraic Riccati equation. Necessary and sufficient conditions for stability of the optimal control and positiveness of Riccati solutions are developed. We show that the concept of weak detectability is not only a sufficient condition for the finiteness of cost functional to imply stability of the associated trajectory, but also a necessary one. This, together with a characterization developed here for the kernel of the Riccati solution, allows us to show that the control solution stabilizes the system if and only if the system is weakly detectable, and that the Riccati solution is positive-definite if and only if the system is weakly observable. The connection between the algebraic Riccati equation and the control problem is made, as far as the minimal positive-semidefinite solution for the algebraic Riccati equation is identified with the optimal solution of the linear quadratic problem. Illustrative numerical examples and comparisons are included.     

Linear matrix inequalities, Riccati equations, and indefinitestochastic linear quadratic controls

This paper deals with an optimal stochastic linear-quadratic (LQ) control problem in infinite time horizon, where the diffusion term in dynamics depends on both the state and the control variables. In contrast to the deterministic case, we allow the control and state weighting matrices in the cost functional to be indefinite. This leads to an indefinite LQ problem, which may still be well posed due to the deep nature of uncertainty involved. The problem gives rise to a stochastic algebraic Riccati equation (SARE), which is, however, fundamentally different from the classical algebraic Riccati equation as a result of the indefinite nature of the LQ problem. To analyze the SARE, we introduce linear matrix inequalities (LMIs) whose feasibility is shown to be equivalent to the solvability of the SARE. Moreover, we develop a computational approach to the SARE via a semi-definite programming associated with the LMIs. Finally, numerical experiments are reported to illustrate the proposed approach     

On the matrix Riccati equation for linear systems with random gain

Considered is the asymptotic property of the discrete-time matrix Riccati equation arising in the optimal control of linear systems with a random gain. The instability and stability conditions are derived in terms of the degree of stability of the state transition matrix.     

On closed form solutions for the differential matrix Riccati equation problem

A new formulation of the differential matrix Riccati equation is presented and a closed analytical solution is obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space. The formulation generalizes the results of [1] on algebraic equations to differential matrix Riccati equations. To illustrate the usefulness of the method, a closed analytical solution of the differential matrix Riccati equation is obtained inR^{2 times 2}.     

Recursive approximate solution to time-varying matrix differential Riccati equation: linear and nonlinear systems

An approximate solution is proposed for linear-time-varying (LTV) systems based on Taylor series expansion in a recursive manner. The intention is to present a fast numerical solution with reduced sampling time in computation. The proposed procedure is implemented on finite-horizon linear and nonlinear optimal control problem. Backward integration (BI) is a well known method to give a solution to finite-horizon optimal control problem. The BI performs a two-round solution: first one elicits an optimal gain and the second one completes the answer. It is very important to finish the backward solution promptly lest in practical work, system should not wait for any action. The proposed recursive solution was applied for mathematical examples as well as a manipulator as a representative of complex nonlinear systems, since path planning is a critical subject solved by optimal control in robotics.     

Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular fuzzy system with cross-term using neural networks

In this paper, solution of generalized matrix Riccati differential equation (GMRDE) for indefinite stochastic linear quadratic singular fuzzy system with cross-term is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of GMRDE obtained from well-known traditional Runge Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of GMRDE is computed by feed forward neural network (FFNN). Accuracy of the solution of the neural network approach to this problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.     

Tutorial solution to the linear quadratic regulator problem using H2 optimal control theory

H ₂ optimal control theory is used to derive a simple solution to the linear quadratic regulator (LQR) problem.     

Bounds on eigenvalues of matrix products with an application to thealgebraic Riccati equation

Lower and upper summation bounds for the eigenvalues of the product XY are presented, under various restrictions on matrices X, Y∈R^n×n. An application to the algebraic Riccati equation yields a trace lower bound. It is observed that these bounds are tighter than those in the literature     

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.     

Iterative matrix bounds and computational solutions to the discretealgebraic Riccati equation

Bilateral matrix bounds for the solution of the discrete algebraic Riccati equation (DARE) are presented. They are new or tighter than the existing bound. Computational algorithms to solve the DARE follow     

Numerical integration of the differential matrix Riccati equation

Two new Bernoulli substitution methods for solving the Riccati differential equation are tested numerically against direct integration of the Riccati equation, the Chandrasekhar algorithm, and the Davison-Maki method on a large set of problems taken from the literature. The first of these new methods was developed for the time-invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric Riecati equation whose solution can be given explicitly in terms of a matrix exponential of ordern. This method is fast and accurate when the extremal solutions of the associated algebraic Riccati equation are well separated. The second new method was developed as a means of eliminating the instabilities associated with the Davison-Maki algorithm. By using reinitialization at each time step the Davison-Maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time-invariant and time-dependent problems. From the results of our study we conclude that the modified Davison-Maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the Chandrasekhar algorithm is better.     

Solution bounds of the continuous Riccati matrix equation

A new approach is proposed for estimating the solution of the continuous algebraic Riccati equation (CARE). Upper and lower solution bounds of the CARE are presented. Comparisons show that the present bounds are more general and/or tighter than existing results.     

Shifting the closed-loop spectrum in the optimal linear quadratic regulator problem for hereditary systems

In the optimal linear quadratic regulator problem for finite-dimensional systems, the method known as an α-shift can be used to produce a closed-loop system whose spectrum lies to the left of some specified vertical line; that is, a closed-loop system with a prescribed degree of stability. This note treats the extension of the α-shift to hereditary systems. As in finite dimensions, the shift can be accomplished by adding α times the identity to the open-loop semigroup generator and then solving an optimal regulator problem. However, this approach does not work with a new approximation scheme for hereditary control problems recently developed by Kappel and Salamon. Since this scheme is among the best to date for the numerical solution of the linear regulator problem for hereditary systems, an alternative method for shifting the closed-loop spectrum is needed. An α-shift technique that can be used with the Kappel-Salamon approximation scheme is developed. A numerical example which demonstrates the feasibility of the method is included.     

A reformulation of the algebraic Riccati equation problem

The algebraic Riccati equation problem is reformulated so as to yield a simple solution when the system has only real roots, as may occur when using a spatially quantized distributed parameter model. A restriction is also placed on the choice of the synthetic output matrixC.     

Partial eigenstructure assignment problem and its application to the constrained linear problem

This article presents a new partial eigenstructure assignment method. This technique keeps the open-loop stable eigenvalues and the corresponding eigenspace unchanged. The remaining undesirable eigenvalues are replaced by other chosen values. This methodology is easy and permits to overcome some limitations encountered in the previous methods. Furthermore, our method is applied to solve the constrained control problem for linear invariants continuous-time systems. Indeed, the problem of finding a stabilising regulator matrix gain taking into account the asymmetrical control constraints is transformed to a Sylvester equation resolution. Examples are given to illustrate the obtained results.     

Robust linear quadratic regulator applied to an inverted pendulum

The natural instability of an inverted pendulum and its dynamics richness, in terms of nonlinearity, provide a nice apparatus to reproduce behaviors of analogous systems. In this way, it is useful to perform benchmark tests for new control approaches developed. In this paper, we address the main inverted pendulum problems: pendulum stabilization, tracking, and catching swing-up control. We show how robust recursive, control and filtering, techniques improve the system performance. They are developed to solve stochastic problems based on deterministic approaches, in order to decrease the worst influence of uncertainties. Experimental results of the proposed robust approach provide robust stability and performance despite parametric uncertainties, disturbances, and noise effects.     

Solution of the matrix Riccati equation in optimal control

This paper is concerned with the solution of the finite-time Riccati equation. The solution to the Riccati equation is given in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and are solved using methods developed in the fields of free vibration theory and aircraft flutter analysis. Simple examples illustrating the method are presented.