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.      

Control strategies on stable manifolds for energy-efficient swing-ups of double pendula

Optimal control problems for mechanical systems often arise in technical applications. To find solutions with minimal control effort, the system’s natural, uncontrolled dynamics can be used. Promising candidates to be considered for energy-efficient trajectories are highly dynamic, but uncontrolled motions on (un)stable manifolds of equilibria. In this contribution, we propose a control strategy for mechanical systems which sequences uncontrolled trajectories on (un)stable manifolds with short control manoeuvres to design a feedforward control. In particular, we present optimal swing-up solutions for a double pendulum which are based on trajectories on the stable manifold of the pendulum’s up–up equilibrium. To demonstrate the advantages of our approach compared to a black-box optimisation, we perform a post-optimisation with the optimal control sequence as an initial guess. The numerical results are evaluated in a simulation environment for the double pendulum on a cart and applied to a real test rig.     

Precise piston trajectory control for a free piston engine

A free piston engine removes the mechanical constraint on the piston motion by eliminating the crankshaft. The extra degree of freedom offers many advantages for reducing fuel consumption and emissions. Nevertheless, stability and robustness of the engine operation has been affected in the meantime. To ensure smooth engine operation, an active motion controller, which utilizes robust repetitive control, was developed previously to regulate the piston motion of a hydraulic free piston engine to track pre-defined trajectories. However, the long piston stroke length, high operating frequency and system nonlinearity impose challenges to precise piston motion control. Therefore, feedforward controllers are investigated in this paper to complement the repetitive control to further improve the tracking performance. The first feedforward design involves the inversion of a linear plant model that describes the dynamics of the engine operation, and the second design is based on the flatness approach, which involves the inversion of a nonlinear model of the system. The two feedforward controllers are designed and implemented on the free piston engine. The experimental and simulation results demonstrate the effectiveness of the proposed control under various operating conditions and reference piston trajectories.     

Feedforward control to track the output of a forced model

In this correspondence a feedforward problem is considered which determines a solution for the state and control trajectories that occur when the output of a plant tracks the output of a model perfectly. The model can have time-varying inputs which are not known in advance. The feedforward control solution is shown to generate a solution to a model matching problem.     

Nonlinear control techniques for an atomic force

Two nonlinear control techniques are proposed for an atomic force microscope system. Initially ,a learning- based control algorithm is developed for the microcantilever-sample system that achieves asymptotic cantilever tip tracking for periodic trajectories. Specifically ,the control approach utilizes a learning- based feedforward term to compensate for periodic dynamics and high- gain terms to account for non-periodic dynamics. An adaptive control algorithm is then developed to achieve asymptotic cantilever tip tracking for bounded tip trajectories despite uncertainty throughout the system parameters. Simulation results are provided to illustrate the efficacy and performance of the control strategies.     

Nonlinear control techniques for an atomic force microscope system

Two nonlinear control techniques are proposed for an atomic force microscope system. Initially, a learning-based control algorithm is developed for the microcantilever-sample system that achieves asymptotic cantilever tip tracking for periodic trajectories. Specifically, the control approach utilizes a learning-based feedforward term to compensate for periodic dynamics and high-gain terms to account for non-periodic dynamics. An adaptive control algorithm is then developed to achieve asymptotic cantilever tip tracking for bounded tip trajectories despite uncertainty throughout the system parameters. Simulation results are provided to illttstrate the efficacy and performance of the control strategies.     

Nonlinear control techniques for an atomic force microscope system

Two nonlinear control techniques are proposed for an atomic force microscope system.Initially,a learning-based control algorithm is developed for the microcantilever-sample system that achieves asymptotic cantilever tip tracking for periodic trajectories.Specifically,the control approach utilizes a learning-based feedforward term to compensate for periodic dynamics and high-gain terms to account for non-periodic dynamics.An adaptive control algorithm is then developed to achieve asymptotic cantilever tip tracking for bounded tip trajectories despite uncertainty throughout the system parameters.Simulation results are provided to illustrate the efficacy and performance of the control strategies.     

Cascaded two-degree-of-freedom control of seeded batch crystallisations based on explicit system inversion

Simple structures and robustness against disturbances are important attributes of chemical productions controllers. The present contribution considers these aspects for seeded batch cooling crystallisations. A new cascaded control scheme is presented. It combines consistent feedforward control with classic feedback control of the main physical process variables, which are the crystalliser temperature, the supersaturation and the crystal mean size. The calculation of the feedforward trajectories uses an explicit inversion of the crystalliser model which is based on the Methods of Moments. A state observer is used to determine online the respective moments of the crystal size distribution. An additional observer is included to estimate unmeasurable heat disturbances and to update the temperature feedforward trajectories. The present contribution summarises the model derivation, system inversion, feedforward and feedback controller design and the design of the observers. Numerical simulations and experimental results from a laboratory plant at BASF Ludwigshafen prove the applicability of the proposed control concept.     

Tracking control for boundary controlled parabolic PDEs with varying parameters: Combining backstepping and differential flatness

The combination of backstepping-based state-feedback control and flatness-based trajectory planning and feedforward control is considered for the design of an exponentially stabilizing tracking controller for a linear diffusion-convection-reaction system with spatially and temporally varying parameters and nonlinear boundary input. For this, in a first step the backstepping transformation is utilized to determine a state-feedback controller, which transforms the original distributed-parameter system into an appropriately chosen exponentially stable distributed-parameter target system of a significantly simpler structure. In a second step, the flatness property of the target system is exploited in order to determine the feedforward controller, which allows us to realize the tracking of suitably prescribed trajectories for the system output. This results in a systematic procedure for the design of an exponentially stabilizing tracking controller for the considered general linear diffusion-convection-reaction system with varying parameters, whose applicability and tracking performance is evaluated in simulation studies.     

Stochastic optimal open-loop feedback control: Approximate solution of the Hamiltonian system

Considering a dynamic control system with random model parameters and using the stochastic Hamilton approach stochastic open-loop feedback controls can be determined by solving a two-point boundary value problem (BVP) that describes the optimal state and costate trajectory. In general an analytical solution of the BVP cannot be found. This paper presents two approaches for approximate solutions, each consisting of two independent approximation stages. One stage consists of an iteration process with linearized BVPs that will terminate when the optimal trajectories are represented. These linearized BVPs are then solved by either approximation fixed-point equations (first approach) or Taylor-Expansions in the underlying stochastic model parameters (second approach). This approximation results in a deterministic linear BVP, which can be handled by solving a matrix Riccati differential equation.     

Model based feedforward control of an industrial glass feeder

This article presents the automation of set point changes of an industrial glass feeder in container glass production. A model is proposed consisting of multiple first order partial differential equations (PDEs). Based on the derived model a feedforward control approach is presented. The approach allows for the calculation of control inputs out of reference trajectories of the system outputs and is used to perform automated set point changes with short transition time. Finally, the approach is implemented at an industrial glass feeder. Measurement results from the Thüringer Behälterglas GmbH (Schleusingen, Germany) are included.     

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.     

Tracking trajectories of feedforward systems

The objective of this paper is to solve the problem of tracking trajectories of feedforward systems. A family of time-varying state feedbacks that globally, uniformly, asymptotically and locally exponentially stabilize trajectories which are not necessarily periodic functions of time is exhibited. The control design is based on the construction of a strict Lyapunov function.     

Inversion-based feedforward control design for linear transport systems with spatially acting input

The considered transport systems, which possess an actuator with a spatial influence characteristic along the transport path, are used for the transport of material between different process stages and for the conditioning of the conveyed goods at the same time. The spatially acting input results in complex input/output behaviour of a Single-Input Single-Output (SISO) system with distributed delays. The feedforward control task under consideration is defined by a setpoint change of the subsequent process stage. For the feedforward controller design, an inversion-based approach in the frequency domain is investigated to steer the output of the transport system towards a predefined constant value. In order to compensate model uncertainties and reject unknown disturbances, the results of the inversion-based feedforward control are used to design a feedback controller.     

Model-based trajectory planning for flexible-link mechanisms with bounded jerk

This paper deals with the model-based development of optimal jerk-limited point-to-point trajectories for flexible-link robotic manipulators. In the proposed approach, an open-loop optimal control strategy is applied to an accurate dynamic model of flexible multi-body planar mechanisms. The model, which has already been fully validated through experimental tests, is based on finite element discretization and accounts for the main geometric and inertial non-linearities of the linkage. Exploiting an indirect variational solution method, the necessary optimality conditions deriving from Pontryagin's minimum principle are imposed, and lead to a differential Two-Point Boundary Value Problem (TPBVP); numerical solution of the latter is accomplished by means of collocation techniques. The resulting motion and control profiles can be used as feedforward reference signals for a position and vibration control. Considering a lightweight RR robot, simulation results are provided for rest-to-rest, jerk-limited trajectories with minimum actuator jerks and vibrations. However, the strategy under investigation has general validity and can be applied to other types of mechanisms, as well as with different objective functions and boundary conditions. Numerical evidence clearly indicates that the use of a composite cost functional and the imposition of jerk constraints can greatly reduce vibration phenomena during high-speed motion of flexible-link manipulators.     

基于观测器的受扰非线性系统近似最优跟踪控制

研究一类受扰非线性系统的最优输出跟踪控制问题.给出了有限时域最优输出跟踪控制律的近似设计算法.首先将求解受扰非线性系统最优跟踪控制问题转换为求解状态向量与伴随向量耦合的非线性两点边值问题,然后利用逐次逼近方法构造序列将其转化为求解两个解耦的线性微分方程序列问题.通过迭代求解伴随向量的序列,可得到由解析的线性前馈-反馈控制部分和伴随向量的极限形式的非线性补偿部分组成的最优输出跟踪控制律.利用参考输入降维观测器和扰动降维观测器,解决了前馈控制的物理可实现问题.最后仿真结果表明了该方法的有效性.     

Two approaches for feedforward control and optimal design of underactuated multibody systems

An underactuated multibody system has less control inputs than degrees of freedom. For trajectory tracking, often a feedforward control is necessary. Two different approaches for feedforward control design are presented. The first approach is based on a coordinate transformation into the nonlinear input–output normal-form. The second approach uses servo-constraints and results in a set of differential algebraic equations. A comparison shows that both feedforward control designs have a similar structure. The analysis of the mechanical design of underactuated multibody systems might show that they are nonminimum phase, i.e., they have unstable internal dynamics. Then the feedforward control cannot be computed by time integration and output trajectory tracking becomes a very challenging task. Therefore, based on the two presented feedforward control design approaches, it is shown that through the use of an optimization procedure underactuated multibody systems can be designed in such a way that they are minimum phase. Thus, feedforward control design using the two approaches is significantly simplified.     

Unconstrained and constrained motion control of a planar two-link structurally flexible robotic manipulator

Unconstrained and constrained motion control of a planar two-link structurally-flexible robotic manipulator are considered in this study. The dynamic model is obtained by using the extended Hamilton's principle and the Galerkin criterion. A method is presented to obtain the linearized equations of motion in Cartesian space for use in designing the control system. The approach to solving the control problem is to use feedforward and feedback control torques. The feedforward torques maneuver the flexible manipulator along a nominal trajectory and the feedback torques minimize any deviations from the nominal trajectory. The feedforward and feedback torques are obtained by solving the inverse dynamics problem for the rigid manipulator and designing linear quadratic Gaussian with loop transfer recovery (LQG/LTR) compensators, respectively. The LQG/LTR design methodology is exploited to design a robust feedback control system that can handle modeling errors and sensor noise, and operate on Cartesian space trajectory errors. Computer simulated results are presented for an example planar, two-link, structurally flexible robotic manipulator. © 1994 John Wiley & Sons, Inc.     

Flatness-based feedforward control for parabolic distributed parameter systems with distributed control

This paper deals with distributed parameter systems being described by inhomogeneous parabolic partial differential equations in one space dimension with a distributed control. The distributed control is presented by the right part of an equation, i.e. the source function. The source function can depend on the time as well as spatial variable. The approach for design of a feedforward control for the purpose of exact output tracking is presented. The design of the feedforward control is based on the examination of inverse system dynamics. The proposed technique utilizes the method of the variables separation and the representation of a solution by the power series in the time domain. Some examples and numerical simulations are included and demonstrate the efficiency of the proposed approach for developing the feedforward control.     

Geometric Methods for Invariant‐Zero Cancellation in Linear Multivariable Systems with Application to Signal Rejection with Preview

This work deals with zero cancellation in linear multivariable systems with possible feedthrough terms from the inputs to the outputs. A methodology for the synthesis of a minimal‐order feedforward compensator preserving key properties of the original system while cancelling minimum‐phase invariant zeros is derived by means of the basic tools of the geometric approach. The properties maintained in the feedforward compensation scheme are stabilizability and right‐invertibility. Duality arguments show how to modify the devised method so as to achieve zero cancellation with a cascade filter retaining detectability and left‐invertibility. Continuous and discrete‐time systems are considered in a unified framework exploiting the common structural features. An original application of zero cancellation to signal rejection with preview is presented. A novel feedforward control scheme is devised, avoiding the steering along minimum‐phase zero techniques that are at the basis of well‐settled solutions.