Knut Graichen

A new approach to inversion-based feedforward control design for nonlinear systems

The finite-time transition between stationary setpoints of nonlinear SISO systems is considered as a scenario for the presentation of a new design approach for inversion-based feedforward control. Design techniques which are based on a stable system inversion result in input trajectories with pre- and/or post-actuation intervals. The presented approach treats the considered transition task as a two-point boundary value problem (BVP) and yields causal feedforward trajectories, which are constant outside the transition interval. The main idea of this approach is to provide free parameters in the desired output trajectory to solve the BVP of the internal dynamics. Thereby, a standard MATLAB function can be used for the numerical solution of the BVP. Feedforward control design techniques are illustrated by simulation results for a simple example.

Brief paper: Swing-up of the double pendulum on a cart by feedforward and feedback control with experimental validation

The swing-up maneuver of the double pendulum on a cart serves to demonstrate a new approach of inversion-based feedforward control design introduced recently. The concept treats the transition task as a nonlinear two-point boundary value problem of the internal dynamics by providing free parameters in the desired output trajectory for the cart position. A feedback control is designed with linear methods to stabilize the swing-up maneuver. The emphasis of the paper is on the experimental realization of the double pendulum swing-up, which reveals the accuracy of the feedforward/feedback control scheme.

Stability and Incremental Improvement of Suboptimal MPC Without Terminal Constraints

The stability of suboptimal model predictive control (MPC) without terminal constraints is investigated for continuous-time nonlinear systems under input constraints. Exponential stability and decay of the optimization error are guaranteed if the number of optimization steps in each sampling instant satisfies a lower bound that depends on the convergence ratio of the underlying optimization algorithm. The decay of the optimization error shows the incremental improvement of the suboptimal MPC scheme.

A Real-Time Gradient Method for Nonlinear Model Predictive Control

Model predictive control (MPC) is a modern control scheme that relies on the solution of an optimal control problem (OCP) on a receding horizon. MPC schemes have been developed in various formulations (regarding continuous/discrete-time systems, finite/infinite horizon length, terminal set/equality constraints, etc.). Comprehensive overviews and references on MPC can, for instance, be found in Diehl et al. (2009); Grune & Pannek (2011); Kothare & Morari (2000); Mayne et al. (2000); Rawlings & Mayne (2009).

Feedforward Control Design for Finite-Time Transition Problems of Nonlinear Systems With Input and Output Constraints

The article extends a recently presented approach to feedforward control design for nonlinear systems to additionally account for input and output constraints. The inversion-based design treats a finite-time transition problem as a two-point boundary value problem (BVP) in the coordinates of the input-output normal form. To account for constraints on the output and its time derivatives, the input-output dynamics is replaced by a new system, which is systematically constructed by means of saturation functions. The solvability of the BVP requires a sufficient number of free parameters in an ansatz function. The resulting BVP with free parameters can be solved in a straightforward manner (e.g., with the Matlab function bvp4c). Input constraints can additionally be considered as constraints on the highest output derivative. The approach is applicable to nonlinear and nonminimum-phase systems, which is illustrated for the side-stepping of an inverted pendulum on a cart.

Trajectory Tracking of a 3DOF Laboratory Helicopter Under Input and State Constraints

This brief deals with the tracking control design of a helicopter laboratory experimental setup. In order to be able to realize highly dynamic flight maneuvers, both input and state constraints have to be systematically accounted for within the control design procedure. The mathematical model being considered constitutes a nonlinear mathematical mechanical system with two control inputs and three degrees of freedom. The control concept consists of an inversion-based feedforward controller for trajectory tracking and a feedback controller for the trajectory error dynamics. The design of the feedforward controller for a point-to-point flight maneuver is traced back to the solution of a 2-point boundary value problem in the Byrnes-Isidori normal form of the mathematical model. By utilizing special saturation functions, the given constraints in the inputs and states can be systematically incorporated in the overall design process. In order to capture model uncertainties and external disturbance, an optimal state feedback controller is designed on the basis of the model linearization along the desired trajectories. The proposed control scheme is implemented in a real-time environment, and the feasibility and the excellent performance are demonstrated by means of experimental results.

Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control

Newmethodsarepresented toaddresstwoissuesin indirectoptimalcontrol:the calculationofastarting pointfor the numerical solution and the consideration of mixed state-input constraints. In the first method, an auxiliary optimal control problem is constructed from a given initial trajectory of the system. Its adjoint variables are simply zero. This auxiliary problem is then used within a homotopy approach to eventually reach the original optimal control problem and the desired optimal solution. The second method concerns the incorporation of mixed stateinput constraints into the dynamics of the considered optimal control problem. It uses saturation functions which strictly satisfy the constraints. In this way, the original constrained optimal control problem is transformed into an unconstrained one with an additional regularization term. The two approaches are derived within a general framework. For sake of illustration, they are applied to the space shuttle reentry problem, which represents a challenging benchmark due to its high numerical sensitivity and the presence of input and heating constraints. The reentryproblemissolvedwithacollocationmethodanddemonstratestheapplicabilityandaccuracyoftheproposed constructive methods.

Handling constraints in optimal control with saturation functions and system extension

A method is presented to systematically transform a general inequality-constrained optimal control problem (OCP) into a new equality-constrained OCP by means of saturation functions. The transformed OCP can be treated more conveniently within the standard calculus of variations compared to the original constrained OCP. In detail, state constraints are substituted by saturation functions and successively constructed dynamical subsystems, which constitute a (dynamical) system extension. The dimension of the subsystems corresponds to the relative degree (or order) of the respective state constraints. These dynamical subsystems are linked to the original dynamics via algebraic coupling equations. The approach results in a new equality-constrained OCP with extended state and input vectors. An additional regularization term is used in the cost to regularize the new OCP with respect to the new inputs. The regularization term has to be successively reduced to approach the original constrained solution. The new OCP can be solved in a convenient manner, since the stationarity conditions are easily determined and exploited. An important aspect of the saturation function formulation is that the constraints cannot be violated during the numerical solution. The approach is illustrated for an extended version of the well-known Goddard problem with thrust and dynamic pressure constraints and using a collocation method for its numerical solution.

Model predictive control of an overhead crane using constraint substitution

This paper describes a real-time model predictive control (MPC) scheme for an overhead crane subject to state and input constraints. The constraints are taken into account by means of a transformation technique which transforms the system dynamics with the corresponding constraints into a new unconstrained representation. This method allows to reformulate the underlying optimal control problem (OCP) of the MPC scheme into an unconstrained counterpart. The unconstrained OCP is then solved by means of a fast MPC algorithm that uses a finite number of iterations per MPC step in order to ensure real-time feasibility. Simulation as well as experimental results demonstrate the computational performance of the MPC scheme.

A fixed-point iteration scheme for real-time model predictive control

A simple model predictive control (MPC) concept for nonlinear systems under input constraints is considered. The presented algorithm takes advantage of an MPC formulation without terminal constraints in order to solve the optimality conditions by a fixed-point iteration scheme that is easy to implement and of algorithmic simplicity. Sufficient conditions for the contraction of the fixed-point iterations are derived. To allow for a real-time implementation within an MPC scheme, a constant number of fixed-point iterations is used in each sampling step and sufficient conditions for asymptotic stability and incremental reduction of the suboptimality are presented.