Meromorphic solutions of difference Riccati equations

Rotational Motion Control by Feedback with Minimum L1-Norm

Max-plus representation for the fundamental solution of the time-varying differential Riccati equation

The paper presents a state-space equation for the two-link and three-link fingers and proposed a solution of finite-time, nonlinear and closed-loop optimal control tracking problem using the state-dependent Riccati equations. The technique involves the use of the solution of the algebraic Riccati equation for the infinite-time case (hence the technique is approximate) and the change of variables that converts a state-dependent, nonlinear, differential Riccati equation (SD-DRE) to a linear differential Lyapunov equation (DLE) which can be solved in analytical form. The effectiveness is validated by using software simulation and hardware experiment.

T HE relative motion of a follower satellite with respect to the leader in a given circular orbit is described by autonomous nonlinear differential equations. The linearized equations around the null solution are known as Hill–Clohessy–Wiltshire (HCW) equations [1–3]. The HCWequations were used by many authors to study rendezvous problems (see [4] and references therein). The Tschauner–Hempel (TH) equations replace the HCW equations, when the orbit of the leader is eccentric [2,5]. Rendezvous problems along an eccentric orbit were studied in [4,5]. The HCW equations possess periodic solutions, which are useful as temporary orbits before mission and for proximity operations such as inspection and repair. The leader–follower formation and reconfiguration problems based on the periodic solutions were studied bymany authors [6–10]. The TH equations also have periodic solutions. They are characterized by Inalhan et al. [11], and the initialization procedure to periodic motion is given. Periodic solutions also follow from the transition matrix of the TH system given byYamanaka and Ankersen [12]. The effects of eccentricity on the shape and size of relative orbits are studied by Sengupta andVadali [13]. Periodic solutions of the TH equations are used for formation flying [14,15] because no control efforts are needed to maintain them. However, their period is fixed and is equal to that of the leader orbit, which would be inconvenient for a quick inspection of the leader. The shape of periodic solutions is irregular compared with that of the HCWequations, which would be undesirable for some missions. In this Note, active formation flying for the TH system is considered, in which the desired relative orbit of the follower is generated by an exosystem. This allows for flexibility of the shape and period of the reference orbit. Typical examples are elliptic relative orbits of the HCWequations with higher frequencies. Formation flying for the TH system with generated reference orbits has not been studied in the literature. To realize such a formation flying, the output regulation theory for linear periodic systems recently given by Ichikawa and Katayama [16] is employed. Output regulation covers tracking and disturbance rejection [17], but the tracking aspect of the theory is used. The regulator equation, which is necessary to achieve output regulation, is a differential equationwith periodic coefficients. The main contribution of this Note is to show that the regulator equation can be solved algebraically if the system is in the controllable canonical form and the observation matrix is of a special form. To achieve asymptotic tracking, stabilizing feedback controls are necessary. They are designed by the differential Riccati equation (DRE) of the linear quadratic regulator theory [18]. To show the effectiveness of this approach, two examples are given. In the first example, the orbit of the leader is assumed to be circular and the HCW equations are considered. The reference orbit of the follower for formation flying is circularwith arbitrary frequency. Starting from a periodic solution of the HCW equations, the follower satellite is asymptotically steered to the reference orbit. In this case, both the regulator equation and theRiccati equation become algebraic. TheL1 norm of the input ΔV and the settling time are given as functions of the weight parameter in the Riccati equation. The L1 norm of the input to maintain the reference orbit for one period is also calculated. These performance indices are further examined by varying the frequency of the reference orbit. The second example is concerned with the TH equations for the elliptic leader orbit. The reference relative orbit of the follower is circular, as in the first example. The regulator equation has periodic coefficients, but it is algebraically solved and explicitly given. The DRE is solved backward and the stabilizing periodic solution is obtained. TheL1 normof the input and the settling time are computed by varying eccentricity and frequency of the reference orbit.

The standard H/sub /spl infin// problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration.

A limitation of the original gain-scheduling approaches is that the closed-loop stability can only be assured when the underlying parameters vary sufficiently slowly. A remedy exists but requires for its implementation the possible solution of asymptotic Riccati equations (ARE) for an infinite number of different parameter values, and the on-line solution of a Riccati differential equation (RDE) with time-varying coefficient matrices. Our method avoids solving the RDE online and instead uses an explicit transient formula that looks up the predetermined solutions of the associated AREs at a finite set of given system operating points. Furthermore, only a finite number of AREs are solved to determine a finite set of controller gains.

Sliding Sector Control (SSC) design method for nonlinear systems is proposed in such a way that the system trajectories are kept around a nonlinear sliding surface which is surrounded by a nonlinear sliding sector. The SSC for the nonlinear system is derived so that the system trajectories are enforced to stay inside the sliding sector for tracking requirements. State-Dependent Differential Riccati Equations (SDDRE) are solved to design the nonlinear sliding surface for the nonlinear dynamical system. Within this context, SSC having nonlinear(or state-dependent) sliding surfaces are used to have a viable solution for the problem formulation. The evolving solutions of the Differential Riccati Equations are used to create the sliding surface which is kept inside the designed sliding sector so that the stability of the nonlinear system is ensured. The proposed SSC method is experimentally tested by applying it to an inner axis of a two axes gimbal system.

Finite-horizon and infinite-horizon linear quadratic optimal control problems: A data-driven Euler scheme

A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvability is {\it equivalent} to the solvability of the indefinite stochastic LQ problem. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation. An example is presented to illustrate the theory developed.

The scattering formulation characterising the effects of and waves in a physical flow process which includes a series of is discussed. Similar concepts can be applied in the linear optimal control problem which includes terminal constraints and the defining linear and quadratic matrix differential equations are shown to be analogous in both cases. The control problem is also shown to include a series of obstacles which can be interconnected according to the properties of the star product which forms the basis of the scattering representation of the general physical system. The scattering matrix of electrical network theory and the associated power relations are also shown to exist analogously in the linear optimal regulator problem. The resulting incident and reflected variables possess similar properties to the scattering variables of network theory and the particular form of scattering matrix is shown to be closely related to the matrix solution of the steady-state Riccati equation. More general energy considerations are extended to the optimal control problem and discussed in terms of inequality properties of a scattering-type matrix and a particular solution of the Riccati equation is shown to be associated with the property defining the state of dissipation in the physical system.

Chang transformation is a decoupling technique for singularly perturbed linear systems. However, for linear slowly time-varying systems, construction of the transformation requires solutions of Differential Riccati Equation (DRE) and Differential Sylvester Equations (DSE). In this paper, through the construction of a contraction mapping for the remainders R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> and R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> , Theorem 1 provides iterative solutions in an interval of the singular perturbation parameter ε for the DRE and DSE, which are key steps for Chang transformation construction. Based on the iterative solutions, the concepts of pth-order approximated Chang transformation, decoupled system, pth-order approximated slow and fast systems are established, thereby facilitating the analysis in subsequent investigation on estimate of the Singular Perturbation Margin (SPM) for Linear Slowly Time-Varying systems and Nonlinear Slowly Time-Varying systems.

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

Generalized partitioned solutions of Riccati equations are presented in terms of forward and backward-time differentiations that are theoretically interesting, computationally attractive, as well as they provide important new interpretations of these results. This approach leads also to important generalizations of previous Riccati solutions such as the Chandrasekhar and the partitioned algorithms. Specifically, it is shown that the generalized partitioned solutions may be given in terms of a generalized Chandrasekhar algorithm. These generalizations pertain to arbitrary initial conditions and time-varying models. Furthermore, based on these partitioned solutions, robust and fast algorithms are obtained for the effective numerical solution of Riccati equations. A particularly effective doubling algorithm is also given for calculating the steady-state solution of time-invariant Riccati equations. The partitioned algorithms are given exactly in terms of a set of elemental solutions which are both simple as well as completely decoupled, and as such computable in either a parallel or serial processing mode. Moreover, the overall solution is given by a simple recursive operation on the elemental solutions.

Generalized Riccati equations arising in stochastic games

Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed.