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.

Elementary observations on a free-end-time problem of minimizing a functional of the form u ?-> integral( tu)(0) L(u(t))dt, over controls t?-> u(t) is an element of U, U a convex, compact neighbourhood of 0 is an element of R-m, are provided in the context of systems (x)overdot = f (x, u) with prescribed initial and terminal states: x(0) = xi and x(t(u)) = 0. Conditions on f and L are identified which are sufficient for existence and continuity of minimizers. For m = 1, U = [-1, 1] and L: v ?-> 1/2 (1 + v(2)) (reflecting a twofold performance objective with equal 2 weight placed on transition time and energy expenditure), an exposition of aspects of the problem is given in the case of integrator chains. Crown Copyright (C) 2022 Published by Elsevier B.V.

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).

A problem of stabilization by feedback of finitely many oscillators via bilinear control action with a common scalar input u is addressed. Under the assumption that the frequencies of the uncontrolled (u = 0) oscillators are distinct, a globally asymptotically stabilizing feedback is developed.

By way of motivation, consider the linear one-dimensional controlled system $$ \dot{x} = ax + u,\quad x\left( 0 \right) = \xi \in {\mathbb{R}}, $$ (6.1)with real parameter a > 0.

In this chapter and with reference to Fig. 3.1, we consider linear systems with input (control) u and output (observation) y.

To motivate a study of asymptotic behaviour of nonlinear systems modelled by ordinary differential equations and differential inclusions, we indicate how such equations/inclusions arise naturally in control of dynamical process by feedback. The concept of control pertains to modifying the behaviour of the process, by manipulation of inputs to the process, in order to achieve some prescribed goal. Fundamental to this is the notion of feedback: a strategy in which the inputs to the process are determined on the basis of concurrent observations on (or outputs from) the process.

We now turn our attention to the initial-value problem for a nonlinear differential equation of the form $$ \dot{x}\left( t \right) = f\left( {t,x\left( t \right)} \right),\quad x\left( \tau \right) = \xi ,\quad \left( {\tau ,\xi } \right) \in J \times G, $$where \( J \subset {\mathbb{R}} \) is an interval, G is a non-empty open subset of \( {\mathbb{R}}^{N} \) and \( f:J \times G \to {\mathbb{R}}^{N} \).

Systems of linear differential equations form the focus of our first line of investigation. In particular, we will develop a theory of existence and uniqueness of solutions of homogeneous initial-value problems of the form $$ \dot{x}(t) = A(t)x(t) $$ , $$ x(\tau ) = \xi , $$ under the assumption that A is piecewise continuous.

This article provides an overview of the circle criterion and its connection with ISS. Classical absolute stability theory and the circle criterion in particular, is concerned with the analysis of a feedback interconnection of Lure type, which consists of a linear system in the forward path and a sector-bounded nonlinearity in the negative feedback path. Classical absolute stability results are revis ited in the context of systems described by differential inclusions and within a framework based on the complex Aizerman conjecture. Contrast with the classical literature that is focused mainly on asymptotic stability of the feedback interconnection, ISS issues are addressed and resolved.

For a class of dissipative nonlinear systems, it is shown that an iISS (integral input-to-state stability) gain can be computed directly from the corresponding supply function. The result is used to prove the convergence to zero of the state whenever the input signal has bounded energy, where the energy functional is determined by the supply function.

The initial-value problem for a class of Volterra functional differential equations- of sufficient generality to encompass, as special cases, ordinary differential equations, retarded differential equations, integro-differential equations, and hysteretic differential equations- is studied. A self-contained and elementary treatment of this over-arching problem is provided, in which a unifying theory of existence, uniqueness, and continuation of solutions is developed. As an illustrative example, a controlled differential equation with hysteresis is considered.

Tracking-by the system output-of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear, single-input, single-output systems modelled by functional differential equations and subject to input saturation. Prespecified is a parameterized performance funnel within which the tracking error is required to evolve; transient and asymptotic behaviour of the tracking error is influenced through choice of parameter values which define the funnel. The control structure is a saturating error feedback with time-varying nonmonotone gain designed to evolve in such a way as to preclude contact with the funnel boundary. A feasibility condition-formulated in bounds of the plant data, the saturation bound, the funnel data, the reference signal, and the initial data-is presented under which the tracking objective is achieved, whilst maintaining boundedness of the state and gain function.

Tracking-by the system output-of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear, single-input, single-output systems modelled by functional differential equations and subject to input saturation. Prespecified is a parameterized performance funnel within which the tracking error is required to evolve; transient and asymptotic behaviour of the tracking error is influenced through choice of parameter values which define the funnel. The control structure is a saturating error feedback with time-varying nonmonotone gain designed to evolve in such a way as to preclude contact with the funnel boundary. A feasibility condition-formulated in bounds of the plant data, the saturation bound, the funnel data, the reference signal, and the initial data-is presented under which the tracking objective is achieved, whilst maintaining boundedness of the state and gain function.

Tracking of reference signals (assumed bounded with essentially bounded derivative) is considered for a class of single-input, single-output, nonlinear systems, described by a functional differential equation with a hysteresis nonlinearity in the input channel. The first control objective is tracking, by the output, with prescribed accuracy: determine a feedback strategy which ensures that, for every reference signal and every system of the underlying class, the tracking error ultimately satisfies the prescribed accuracy requirements. The second objective is guaranteed output transient performance: the graph of the tracking error should be contained in a prescribed set (performance funnel). Under a weak sector boundedness assumption on the hysteresis operator, both objectives are achieved by a memoryless feedback which is universal for the underlying class of systems.

Tracking of an absolutely continuous reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of non-linear, single-input, single-output, dynamical systems modelled by functional differential equations satisfying certain structural hypotheses (which, interpreted in the highly specialised case of linear systems, translate into assumptions of (i) relative degree one, (ii) positive high-frequency gain and (iii) stable zero dynamics). The control objective is evolution of the tracking error within a prespecified funnel, thereby guaranteeing prescribed transient performance and prescribed asymptotic tracking accuracy. This objective is achieved by a control which takes the form of linear error feedback with time-varying gain. The gain is generated by a non-linear feedback law in which the reciprocal of the distance of the tracking error to the funnel boundary plays a central role. In common with many established adaptive control methodologies, the overall feedback structure exploits an intrinsic high-gain property of the system, but differs from these methodologies in two fundamental respects: the funnel control gain is not dynamically generated and is not necessarily monotone. The main distinguishing feature of the present article vis à vis its various precursors is twofold: (a) non-linearities of a general nature can be tolerated in the input channel; (b) a more general formulation of prescribed transient behaviour is encompassed (including, for example, practical (M, μ)-stability wherein, for prescribed parameter values M > 1, μ > 0 and λ > 0, the tracking error e(·) is required to satisfy |e(t)| < max {Me −μt |e(0)|, λ} for all t ≥ 0).

Results of input-to-state stability (ISS) type for systems of Lur'e type are presented. The generic system is a feedback interconnection of a finite-dimensional linear system and a set-valued nonlinearity. An illustrative example is provided wherein the results are applied in the design of PID control in the presence of input quantization and nonlinearity.

Input-to-state stability (ISS) of a class of differential inclusions is proved. Every system in the class is of Lur'e type: a feedback interconnection of a linear system and a set-valued nonlinearity. Applications of the ISS results, in the context of feedback interconnections with a hysteresis operator or a quantization operator in the feedback path, are developed.