Anna Prach

A numerical comparison of frozen-time and forward-propagating Riccati equations for stabilization of periodically time-varying systems

Feedback control of linear time-varying systems arises in numerous applications. In this paper we numerically investigate and compare the performance of two heuristic techniques. The first technique is the frozen-time Riccati equation, which is analogous to the state-dependent Riccati equation, where the instantaneous dynamics matrix is used within an algebraic Riccati equation solved at each time step. The second technique is the forward-propagating Riccati equation, which solves the differential algebraic Riccati equation forward in time rather than backward in time as in optimal control. Both techniques are heuristic and suboptimal in the sense that neither stability nor optimal performance is guaranteed. To assess the performance of these methods, we construct Pareto efficiency curves that illustrate the state and control cost tradeoffs. Three examples involving periodically time-varying dynamics are considered, including a second-order exponentially unstable Mathieu equation, a fourth-order rotating disk with rigid body unstable modes, and a 10th-order parametrically forced beam with exponentially unstable dynamics. The first two examples assume full-state feedback, while the last example uses a scalar displacement measurement with state estimation performed by a dual Riccati technique.

Development of a State Dependent Ricatti Equation Based Tracking Flight Controller for an Unmanned Aircraft

A dual loop nonlinear State Dependent Riccati Equation (SDRE) control method is developed for the ight control of an unmanned aircraft. The outer loop addresses the attitude and altitude kinematics, while the inner loop handles the translational and rotational equations of motion. The control strategy utilizes a tracking control problem. The mismatch due to the SDC factorization of the inner loop is handled with a nonlinear compensator again derived from the tracking control formulation. The quadratic optimal control problems of the inner and outer loops are solved at discrete intervals in time. A nonlinear simulation model of the UAV is used to examine the performance of the SDRE controller. Two ight scenarios are considered: a coordinated turn maneuver and a high angle of attack ight. These simulation results show the eectiveness of the proposed nonlinear controller.

Infinite-Horizon Linear-Quadratic Control by Forward Propagation of the Differential Riccati Equation [Lecture Notes]

One of the foundational principles of optimal control theory is that optimal control laws are propagated backward in time. For linear-quadratic control, this means that the solution of the Riccati equation must be obtained from backward integration from a final-time condition. These features are a direct consequence of the transversality conditions of optimal control, which imply that a free final state corresponds to a fixed final adjoint state [1], [2]. In addition, the principle of dynamic programming and the associated Hamilton-Jacobi-Bellman equation is an inherently backward-propagating methodology [3].

Faux-Riccati synthesis of nonlinear observer-based compensators for discrete-time nonlinear systems

Pseudo-linear models of nonlinear systems use either a state-dependent coefficient or the Jacobian of the vector field to facilitate the use of Riccati techniques. In this paper we use the state-dependent Riccati equation (SDRE) and the forward propagating Riccati equation (FPRE) with pseudo-linear models to construct nonlinear observer-based compensators for output-feedback control of nonlinear discrete-time systems. While attractive due to their simplicity and potentially wide applicability, these techniques remain largely heuristic. The goal of this paper is thus to present numerical experiments to assess the performance of these “faux” Riccati techniques on representative nonlinear systems. The goal is to compare the performance of SDRE and FPRE when used with either a state-dependent coefficient or the Jacobian of the vector field. Stabilization and performance are considered, along with integral control for step command following.

Adaptive trim and trajectory following for a tilt-rotor tricopter

We apply adaptive control to an unconventional aircraft, namely, a three-rotor flight vehicle, one of whose rotors can tilt about the longitudinal axis of the fuselage. This combination of actuators has aerodynamic advantages but also poses challenges in terms of trimming the aircraft in order to balance the torque about the roll, pitch, and yaw axes. The paper uses retrospective cost adaptive control (RCAC) to trim the aircraft in hover as well as to follow straight-line and circular flight trajectories.

Output-feedback control of linear time-varying and nonlinear systems using the forward propagating Riccati equation:

For output-feedback control of linear time-varying (LTV) and nonlinear systems, this paper focuses on control based on the forward propagating Riccati equation (FPRE). FPRE control uses dual difference (or differential) Riccati equations that are solved forward in time. Unlike the standard regulator Riccati equation, which propagates backward in time, forward propagation facilitates output-feedback control of both LTV and nonlinear systems expressed in terms of a state-dependent coefficient (SDC). To investigate the strengths and weaknesses of this approach, this paper considers several nonlinear systems under full-state-feedback and output-feedback control. The internal model principle is used to follow and reject step, ramp, and harmonic commands and disturbances. The Mathieu equation, Van der Pol oscillator, rotational-translational actuator, and ball and beam are considered. All examples are considered in discrete time in order to remove the effect of integration accuracy. The performance of FPRE is investigated numerically by considering the effect of state and control weightings, the initial conditions of the difference Riccati equations, the domain of attraction, and the choice of SDC.

Nonlinear Aircraft Flight Control Using the Forward Propagating Riccati Equation

This paper presents application of the forward-propagating Riccati equation (FPRE) control for the aircraft flight control system for command following and disturbance rejection. Unlike classical finite-horizon optimal control, where the differential Riccati equation is integrated backwards in time for a given final-time condition, FPRE control uses differential equations that are integrated forward in time. Although this technique is heuristic and guarantees neither performance nor stability, simplicity of the FPRE algorithm makes it attractive for applications for nonlinear systems defined using a state-dependent coefficient parameterization. The performance of the proposed flight control system is investigated via the numerical simulations using a nonlinear model of a fixed-wing aircraft.

Design of a Full State SDRE Tracking Controller for an Unmanned Aircraft

This paper addresses the design of a nonlinear tracking controller for an unmanned aircraft. A State Dependent Riccati Equation (SDRE) method is utilized in a parallel-loop controller structure. We consider a classical approach of separating dynamics of a fixedwing aircraft into longitudinal and lateral. SDRE controllers are designed for each of these control channels. The coupling effects, and nonlinearities are reflected in the extended parametrization models of the channels. A nonlinear simulation model of an aircraft is used to validate the performance of the proposed control algorithm. Simulation results show the effectiveness of the algorithm for a coordinated turn maneuver.