In the context of linear control systems, a commonly-held intuition is that negative and positive feedback cannot both be stability enhancing. The canonical linear prototype is the scalar system x(.) = u which, under negative linear feedback u = -kx (k> 0) is exponentially stable for all k > 0, whereas the lack of exponential instability of the (marginally stable) uncontrolled system is amplified by positive feedback u = kx (k> 0). By contrast, for nonlinear systems it is shown, by example, that this intuitive dichotomy may fail to hold.

The methodology of funnel control was introduced in the early 2000s, and it has developed since then in many respects achieving a level of mathematical maturity balanced by practical applications. Its fundamental tenet is the attainment of prescribed transient and asymptotic behaviour for continuous-time controlled dynamical processes encompassing linear and nonlinear systems described by functional differential equations, differential-algebraic systems, and partial differential equations. Considered are classes of systems specified by structural properties - such as relative degree and stable internal dynamics - of the systems only, the precise systems' data are in general unknown; the latter reflects the property that in general any model of a dynamical process is not precise. Prespecified are: a funnel shaped through the choice of a smooth function and freely chosen by the designer, a fairly large class of smooth reference signals, and a system class satisfying certain structural properties. The aim is to design, based on the structural assumptions and the input and output information only, a single `simple' control strategy -- called the funnel controller -- so that its application to any system of the given class and to any reference signal results in feasibility of the funnel control objective: that is solutions of the closed-loop system do not exhibit blow-up in finite time, all variables are bounded, and -- most importantly -- the evolution of the error between the system's output and the reference signal remains within the prespecified funnel. In the Introduction, we describe the genesis of funnel control. After that, we investigate diverse system classes amenable to funnel control. Funnel control is shown for systems with arbitrary relative degree and systems described by partial differential equations. Finally, we discuss input constraints and applications.

The present work is a successor of [Ilchmann, Kirchhoff 2022] on generic controllability and of [Ilchmann, Kirchhoff 2023] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive new results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices (E,A,B)∈ℝ^ℓ××ℝ^ℓ× n×ℝ^ℓ× m , while for relative genericity we allow the restricted set Σ _ℓ ,n,m^≤ r := {(E,A,B)∈ℝ^ℓ× n×ℝ^ℓ× n×ℝ^ℓ× m | rk _ℝ E ≤ r} , where r∈ℕ . Our main results are characterizations of generic controllability and generic stabilizability in Σ _ℓ ,n,m^≤ r in terms of the numbers ℓ , n, m, r .

A general procedure for recognising and identifying oval forms is suggested and performed on an eighteenth century engraving representing the Bibliotheca Wolfenbüttel from 1705. The importance of finding a systematic, consistent and historically plausible procedure is stressed. It is suggested that research on the subject often lacks insight in what the architect could or may have drawn when planning a building or part of it, leading to possible mistakes in the determination of the form. In this paper both cultural and practical aspects are taken into account.

We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example, state space transformations, invertible transformations from the left and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold.

We study model predictive control for singular differential-algebraic equations with higher index. This is a novelty when compared to the literature where only regular differential-algebraic equations with additional assumptions on the index and/or controllability are considered. By regularisation techniques, we are able to derive an equivalent optimal control problem for an ordinary differential equation to which well-known model predictive control techniques can be applied. This allows the construction of terminal constraints and costs such that the origin is asymptotically stable w.r.t. the resulting closed-loop system.

We show that Funnel MPC, a novel model predictive control (MPC) scheme, allows tracking of smooth reference signals with prescribed performance for nonlinear multi-input multi-output systems of relative degree one with stable internal dynamics. The optimal control problem solved in each iteration of funnel MPC resembles the basic idea of penalty methods used in optimization. To this end, we present a new stage cost design to mimic the high-gain idea of (adaptive) funnel control. We rigorously show initial and recursive feasibility of funnel MPC without imposing terminal conditions or other requirements like a sufficiently long prediction horizon.

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. https://doi.org/10.1007/978-3-642-34928-7_1 )), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.

Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown "control direction" and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal, the tracking error evolves in a prescribed funnel chosen to reflect transient and asymptotic accuracy objectives. Two fundamental properties underpin the system class: bounded-input bounded-output stable internal dynamics, and a high-gain property (an antecedent of which is the concept of sign-definite high-frequency gain in the context of linear systems).

We show that Funnel MPC, a novel Model Predictive Control (MPC) scheme, allows tracking of smooth reference signals with prescribed performance for nonlinear multi-input multi-output systems of relative degree one with stable internal dynamics. The optimal control problem solved in each iteration of Funnel MPC resembles the basic idea of penalty methods used in optimization. To this end, we present a new stage cost design to mimic the high-gain idea of (adaptive) funnel control. We rigorously show initial and recursive feasibility of Funnel MPC without imposing terminal conditions or other requirements like a sufficiently long prediction horizon.

SummaryWe study the optimal control problem (OCP) for regular linear differentialalgebraic systems. To this end, we introduce the input index, which allows, on the one hand, to characterize the space of consistent initial values in terms of a Kalman‐like matrix and, on the other hand, the necessary smoothness properties of the control. The latter is essential to make the problem accessible from a numerical point of view. Moreover, we derive an augmented system as the key to analyze the OCP with tools well known from optimal control of ordinary differential equations. The new concepts of the input index and the augmented system provide easily checkable sufficient conditions, which ensure that the stage costs are consistent with the differential‐algebraic system.

We consider model predictive control (MPC) without stabilizing terminal constraints and costs for systems governed by linear Differential-Algebraic Equations. To this end, an augmented system is introduced to derive an equivalent formulation of the underlying Optimal Control Problem to be solved in each MPC iteration, which is only constrained by an Ordinary Differential Equation. This facilitates the analysis and the computation of a prediction horizon such that asymptotic stability of the origin w.r.t. the MPC closed-loop is guaranteed.

The topic of the master thesis is linear-quadratic optimal control of time-varying and timeinvariant differential-algebraic equations (DAEs). The thesis is divided into two main parts: in the first part, we investigate linear time-varying DAEs. We recall the solution theory of DAEs and introduce the optimal control problem. We then proceed to prove that the optimal value is a quadratic function and fulfils Bellman’s principle of optimality. Using these results, we can characterize the optimal value as an extremal solution of the Kalman-Yakubovich-Popov inequality. In the second part, we turn our attention towards time-invariant, regular DAEs. We first derive a differentiability condition that the control input of the system needs to fulfil. Using these results, we introduce an augmented system that includes certain derivatives of the input as system states. For this augmented system, an optimal control problem equivalent to the one of the nominal system is defined that can be solved easily using results from the theory of optimal control for ordinary differential equations. This enables us to explicitly calculate the optimal control of the nominal DAE, which can be implemented as a state feedback as well.