Jet-scheduling control for flat systems

The control of cranes and nonholonomic robots has gained increased interest mainly because of the civilian and military industrial need to achieve fast and accurate transport of goods and equipment. Old and new harbors are now venturing into fully automated systems combining automated trolleys and classical cranes. From a theoretical viewpoint these systems are challenging because they are strongly dynamically coupled and offer interesting and useful control problems. Therefore, to take full advantage of their potential, the control design must take into account as much structural information as possible. The structural property that is exploited in the control design proposed in this thesis is the differential flatness property of these systems, that is the existence of particular functions of the states (called flat outputs), the time parametrization of which implies parametrization of all the individual states and inputs. This property is extremely useful for motion planning problems where the system should move quickly from one configuration to another, without inducing too much overshoot or residual oscillations. However, the flatness property is not sufficient to guarantee the design of an efficient controller in the presence of uncertain and unmodeled dynamics. This is especially the case for cranes where the winching mechanism, expressed in terms of the engine and pulleys, has a large amount of unmodeled dry friction. This robustness issue is normally addressed by splitting the control task into a feedforward-like part that handles the dynamical couplings and a feedback term that enforces the tracking of the reference values stemming from the feedforward motion planning algorithm. In contrast, this thesis proposes to combine these two mechanisms, resulting in what will be called the jet-scheduling controller. Classically, the flatness property guarantees the construction of a feedforward input based on a planned motion of the flat outputs by simply combining values of the flat outputs and their time derivatives, i.e. without having to integrate differential equations. Therefore, in the absence of perturbation, this mechanism is sufficient to move the system from one state to another, once a trajectory compatible with the initial and final positions has been designed. However, when the system has some unmodeled dynamics, an additional mechanism must be provided to make sure that the planned trajectory is indeed tracked accurately. The point of view adopted in this thesis is that, instead of specifying a trajectory to be tracked explicitly, a dynamical system called "the jet scheduler" provides the derivatives (the jets) of an ideal stabilizing trajectory. These jets are updated regularly according to measurements so as to react to unknown perturbations. The flat correspondence is used to provide the values of the jets, and a subsidiary controller is designed to ensure that these jets are really matched asymptotically by the true system. Unfortunately, each of these mechanisms could possibly break the equivalence between the original nonlinear system and the linear extended system (contrary to the classical feedback linearization approach for which this correspondence is guaranteed at every time instant). The design of the jet-scheduling controllers and the implication of the possible loss of correspondence are detailed in this work. In addition, stability issues are addressed. Applications to two classes of systems are shown, namely, nonholonomic robots and cranes. The specific properties of these systems are used to achieve a rigorous stability proof. The controller for both the nonholonomic robot and a new crane design labeled SpiderCrane that fully takes advantage of the jet-scheduling mechanism are tested on real setups. Nonlinear Control ; Flatness-based Control ; Trajectory Tracking ; Stabilization ; Nonholonomic Robot ; Crane.

Finite Dimensional Methods for Differential Flatness

Basile Graf

In Control System Theory, the study of continuous-time, finite dimensional, underdetermined systems of ordinary differential equations is an important topic. Classification of systems in different categories is a natural initial step to the analysis of a given control problem. Systems of equations can often be “transformed” into other “equivalent” ones. Then, a control system is associated to a set of equivalent equations. In this setting, a property of a control system can be defined as a property that has to be satisfied by some arbitrary system of equations in the set representing the control system. Assessing such a property can be a difficult task. In this thesis, we review and study a number of ways to determine whether a multi-input nonlinear system is flat, i.e. whether it is equivalent to a linear system after some dynamical extension and change of coordinates. This is a difficult as well as a well studied problem. Therefore, coming up with some altogether new approach or solution is to a certain extent illusory. A substantial part of the text is devoted to describing existing approaches and sometimes to propose either an original alternative or an original point of view. Another part of the thesis is dedicated to the study of a drastically reduced version of the problem, where more can be said in an algorithmic way. Nonlinear control systems are first modeled as the embedding of some fibered bundle to the first jet of the time-and-state-variables manifold. The exterior system, or Pfaffian system, corresponding to the ODE then arises naturally. Input prolongations are then introduced as lifts of the previously mentioned embedding. Various filtration techniques and their applications to static feedback linearization are discussed next. A full chapter is devoted to the infinite dimensional approach involving matrix differential operators. A now classical theorem, linking integrability of the basis of a differential module after application of one such an operator, and the flatness property is discussed. The relations obtained can be decomposed in an equivalent set of differentially closed equations. We state a version of the resulting theorem where the “curvature equations” are trivial. A subproblem that has attracted the attention of researchers is the question whether a given system — subject to some state constraints — is flat. In this setting, useful concepts are that of a covering of a system by another one and the accompanying result stating that a flat system can cover only a flat system. Hence, if a “large” linear system is given together with a set of nonlinear constraints, flatness of the constrained system is assessed if the unconstrained system can be shown to cover the constrained one. Starting with the classical notion of controlled invariance and a generalized notion coined dynamic controlled invariance, sufficient conditions are discussed which also involve the notion of right invertibility and the dynamic extension algorithm. Modeling of mechanical systems by free moving point masses subject to some control forces and quadratic constraints is often effective. The resulting unconstrained equations are linear and bilinear in the state and control/Lagrangian multiplier variables. We propose a relative derived flag that leads to a filtration with guaranteed integrability at each stage. This leads to a very effective sufficient condition for the flatness of the unconstrained model. The algorithm, together with the test described in the previous section, is used to show flatness of some generalized pendulum-like equations. They are also shown to specialize to the non-holonomic car equations and to the VTOL/pendulum equations when some specific parameters are suitably chosen.

EPFL2013

A Data-Driven Frequency-Domain Approach for Robust Controller Design via Convex Optimization

Achille Nicoletti

The objective of this dissertation is to develop data-driven frequency-domain methods for designing robust controllers through the use of convex optimization algorithms. Many of today's industrial processes are becoming more complex, and modeling accurate physical models for these plants using first principles may be impossible. With the increased developments in the computing world, large amounts of measured data can be easily collected and stored for processing purposes. Data can also be collected and used in an on-line fashion. Thus it would be very sensible to make full use of this data for controller design, performance evaluation, and stability analysis. The design methods imposed in this work ensure that the dynamics of a system are captured in an experiment and avoids the problem of unmodeled dynamics associated with parametric models. The devised methods consider robust designs for both linear-time-invariant (LTI) single-input-single-output (SISO) systems and certain classes of nonlinear systems. In this dissertation, a data-driven approach using the frequency response function of a system is proposed for designing robust controllers with H∞ performance. Necessary and sufficient conditions are derived for obtaining H∞ performance while guaranteeing the closed-loop stability of a system. A convex optimization algorithm is implemented to obtain the controller parameters which ensure system robustness; the controller is robust with respect to the frequency-dependent uncertainties of the frequency response function. For a certain class of nonlinearities, the proposed method can be used to obtain a best-linear-approximation with an associated frequency dependent uncertainty to guarantee the stability and performance for the underlying linear system that is subject to nonlinear distortions. The concepts behind these design methods are then used to devise necessary and sufficient conditions for ensuring the closed-loop stability of systems with sector-bounded nonlinearities. The conditions are simple convex feasibility constraints which can be used to stabilize systems with multi-model uncertainty. Additionally, a method is proposed for obtaining H∞ performance for an approximate model (i.e., describing function) of a sector-bounded nonlinearity. This work also proposes several data-driven methods for designing robust fixed-structure controllers with H∞ performance. One method considers the solution to a non-convex problem, while another method convexifies the problem and implements an iterative algorithm to obtain the local solution (which can also consider H2 performance). The effectiveness of the proposed method(s) is illustrated by considering several case studies that require robust controllers for achieving the desired performance. The main applicative work in this dissertation is with respect to a power converter control system at the European Organization for Nuclear Research (CERN) (which is used to control the current in a magnet to produce the desired field in controlling particle trajectories in accelerators). The proposed design methods are implemented in order to satisfy the challenging performance specifications set by the application while guaranteeing the system stability and robustness using data-driven design strategies.

EPFL2018

Fixed-Order H-infinity Controller Design via Convex Optimization Using an Alternative to Youla Prameterization

Alireza Karimi, Hamid Khatibi

All H-infinity controllers of a SISO LTI system are parameterized thanks to the relation between Bounded Real Lemma and Positive Real Lemma and a new concept of strict positive realness of two transfer functions with the same Lyapunov matrix in the matrix inequality of the Kalman-Yakubovic-Popov lemma. This new parameterization shares the same features with Youla parameterization, namely on the convexity of H-infinity norm constraints for the closed-loop transfer functions. However, in contrary to Youla parameterization, it can deal with any controller order and any controller structure such as e.g. PID. The main feature of the proposed method is that it can be extended easily for the systems with polytopic uncertainty. This way, a convex inner approximation of all H-infinity controllers for polytopic systems is given, which can be enlarged by increasing the controller order. In order to design a low-order robust H-infinity controller with less conservatism, rank of the k-th Sylvester resultant matrix of the controller is made to be deficient via a convex approximation of the rank minimization problem. The effectiveness of the proposed method is shown via simulation results.