Exact output tracking in prescribed finite time via funnel control

Output reference tracking of unknown nonlinear systems is considered. The control objective is exact tracking in predefined finite time, while in the transient phase the tracking error evolves within a prescribed boundary. To achieve this, a novel high-gain feedback controller is developed that is similar to, but extends, existing high-gain feedback controllers. Feasibility and functioning of the proposed controller is proven rigorously. Examples for the particular control objective under consideration are, for instance, linking up two train sections, or docking of spaceships.


Introduction
The control objective, to bring the output of a system to a certain exact value within predefined finite time has various applications: in modern robot-based industry (placement of components), in public transportation (connection of two train sections), autonomous driving (docking at the charging station), and in astronautics (rendezvous of spacecraft), to name but a few.While asymptotic exact tracking has been studied for some time, there are few results on exact tracking in finite time.In [14], referring to the results in [6], it is shown that the proposed funnel controller achieves global asymptotic stabilization for a class of linear multi-input multi-output (MIMO) systems of relative degree one.A generalization to a class of nonlinear relative degree one MIMO systems is proposed in [26].In [24] an extended sliding mode controller is proposed, which achieves asymptotic tracking of linear single-input single-output (SISO) systems.This controller is extended to linear MIMO systems in [23].In [8] backstepping is combined with feedback linearization techniques and higher order sliding modes to design a controller, which achieves exponential tracking.In [25] a high-gain based slid-⋆ This work was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft); Project-IDs: 362536361, and 471539468.
ing mode controller is introduced, where the peaking related to the high-gain observer is avoided by introducing a dwell-time activation scheme.This controller achieves asymptotic tracking for a class of nonlinear SISO systems of arbitrary relative degree, where the reference signal is generated by a reference model.To the price of a discontinuous control, asymptotic tracking for nonlinear MIMO systems is achieved in [30,31].In [19] a funnel controller is proposed, which achieves asymptotic tracking for a class of nonlinear relative degree one MIMO systems.This result is extended in the recent work [3], where it is shown that the proposed controller achieves asymptotic tracking of nonlinear MIMO systems with arbitrary relative degree whereas the tracking error has prescribed transient behaviour.We now turn from asymptotic tracking towards exact tracking in finite time.In [1] a recursive observer structure as well as an extension of the homogeneous approximation technique is introduced to achieve global asymptotic as well as finite time stabilization for higher order chain of integrator systems.For control affine systems with given relative degree, in [21] a homogeneous higher order sliding mode controller is designed, which achieves stabilization in finite time.In [2] sliding mode control concepts and results from [20,4] are used to establish control schemes, which achieve finite time stabilization for linear SISO & MIMO systems.In [11] backstepping and higher order sliding mode control are combined to construct a controller, which achieves exact output tracking in finite time for nonlinear MIMO systems in nonlinear block controllable form.Similar to the prescribed performance controller in [9], this controller suffers from the proper initialization problem, where it is not clear, how large to choose the involved parameters.The controller in [11], along with limiting conditions on the system class, presumes knowledge of the system's functions and explicitly involves inverses of some.In [32] a controller is introduced, which achieves exact tracking in finite time for a class of nonlinear SISO systems satisfying a certain homogeneity assumption.This controller relies on estimations of the external disturbances, where the problem of proper initialization is avoided by assuming explicit knowledge of the bounds of the disturbances and the reference.The controller explicitly involves (parts of) the system's right hand side and is of relatively high complexity.Utilizing the implicit Lyapunov function approach, in [22] a state feedback-integral controller is designed, which is capable to stabilize homogeneous systems (negative and positive) in fixed finite time, where the final time can be estimated involving the initial state and a corresponding Lyapunov function.In most of the control schemes for exact tracking in finite time discussed above, the final time cannot be prescribed; only the existence of such a finite time is ensured.Contrary, in [16] a controller is introduced which solves a predefined-time exact tracking problem for the class of fully actuated mechanical (relative degree two) systems.The controller relies on a backstepping procedure and consists of a predefinedtime stabilization function, and involves the system's equations explicitly.In [29] a funnel controller is introduced, which achieves asymptotic tracking as well as convergence to zero of the tracking error in finite time for a class of relative degree one SISO systems.Regularization in prescribed finite time is achieved in [28] for nonlinear systems in strict feedback form, invoking a strictly increasing scaling function for the state, which grows unbounded when approaching the final time.In the recent work [10], linear time invariant systems with delayed input are under consideration.Representing the delay system in a PDE-ODE cascade, under the usage of backstepping techniques and integral transformations a controller is designed, which stabilizes the system within predefined finite time.Circumventing some drawbacks mentioned above, we propose a controller, which achieves exact tracking in predefined finite time.Since the controller is of funnel type it inherits the advantages of robustness with respect to noise, and that the tracking error evolves within prescribed bounds.Moreover, the controller is model-free in the sense that no knowledge of the system's parameters is assumed; only knowledge of the order r ∈ N of the differential equation and the common dimension m ∈ N of the input and output is required, and the system's right-hand side has to satisfy a highgain property.The system class under consideration is the same as in [3], and encompasses the systems under consideration in [1,21,2,11,16,29,22], and under additional regularity assumptions those in [32].
As the main contribution we develop a feedback controller, which is designed to achieve satisfaction of a particular control objective.While the recently proposed funnel controller [3] achieves asymptotic exact tracking, the controller in the present article achieves exact tracking in predefined finite time.The latter means that the output y of a system is forced to approach a given reference y ref , and lim t→T y(t) = y ref (T ) for a predefined final time T , cf. Figure 1.The result in the present article closes a gap in the existing theory of funnel control, and is formulated in Theorem 3.1.We rigorously prove feasibility of the proposed controller.To this end, we extend and generalize the existing feasibility proof [3].Specifically, the proof in [3] extensively uses a growth condition on the involved "funnel function" φ, while in the present article the respective function is unbounded, it even has a pole.Since the proof is quite technical, it is relegated to the appendix.

Control objective, system class, feedback law
We state the problem under consideration, introduce the class of systems to be controlled, and define the feedback law.To convey the basic idea, we briefly give the general framework and then present the individual components in detail in Sections 2.2 and 2.3.We consider multi-input multi-output r th -order functional differential equations y (r) (t) = f d(t), T(y, ẏ, . . ., y (r−1) )(t), u(t) , with bounded unknown disturbance d, unknown nonlinear function f and unknown operator T, the latter are characterized in Definitions 2.1 and 2.2 below.If σ > 0, then the initial value is given via the initial trajectory y 0 ; if σ = 0, then the initial value is given by (y(0), ẏ(0), . . ., y (r−1) (0)) ∈ R rm .Beside typical physical phenomena such as, e.g., hysteresis effects, the operator T can also model delay elements, cf.[14,Sec. 4.4].
If delays are involved, σ > 0 corresponds to the initial delay.Note that the input u and the output y have the same dimension m ∈ N.
is the group of invertible R n×n matrices; for I ⊆ R an interval L ∞ loc (I; R p ) is the set of locally essentially bounded functions f :  ) such that x| [0,ω) is absolutely continuous and satisfies ẋi (t) = x i+1 (t) for i = 1, . . ., r − 1, and ẋr (t) = f (d(t), T(x(t)), u(t)) (which corresponds to (1)) for almost all t ∈ [0, ω).A solution x is said to be maximal, if it does not have a right extension which is also a solution.

Control objective
We aim to design a feedback controller, which achieves exact reference tracking in the following sense.For a given reference trajectory y ref ∈ W r,∞ ([0, T ); R m ) and a predefined final time T > 0, the output y of system (1) approaches the reference within the interval [0, T ), and coincides with the reference as t → T , i.e., for e(•) where e (i) (•) denotes the i th derivative of e(•).Moreover, in the transient phase for t ∈ [0, T ) the error evolves within the so called "performance funnel", i.e., (t, e(t)) where φ is a boundary function defined in (4a).The control objective is illustrated in Figure 1.

System class
To introduce the system class under consideration, we first provide some necessary definitions.To characterize the class of admissible nonlinearities f in system (1), we recall the definition of the "high-gain property" from [3, Sec.1.2].
Definition 2.1.For p, q, m ∈ N a function f ∈ C(R p × R q × R m ; R m ) satisfies the high-gain property, if there exists ρ ∈ (0, 1) such that, for every compact K p ⊂ R p and compact K q ⊂ R q the continuous function In Remark 2.5 we discuss the high-gain property in detail.The operator T in (1) belongs to the operator class defined below.This definition is taken from [3, Sec.1.2].
Definition 2.2.If for n, q ∈ N and σ ≥ 0 the operator (a) T maps bounded trajectories to bounded trajectories, i.e., for all c 1 > 0, there exists (c) T is locally Lipschitz continuous in the following sense: for all t ≥ 0 and all ξ then the operator T belongs to the operator class T n,q σ .
With the definitions so far, we may introduce the system class under consideration, which is the same as in [3].
Definition 2.3.For m, r ∈ N a system (1) is said to belong to the system class N m,r , if for some p, q ∈ N the "disturbance" is bounded, i.e., d ) satisfies the highgain property from Definition 2.1 and for σ ≥ 0 the operator T belongs to T rm,q σ ; we write (d, f, T) ∈ N m,r .
Remark 2.4.For n ∈ N, consider a state-space model where y(t) ∈ R m is the output, and for m ≤ n, f : R n → R n , g : R n → R n×m , h : R n → R m sufficiently smooth.
Assume this system has relative degree (r 1 , . . ., r m ) = (r, . . ., r) ∈ N m for r ∈ N at x 0 ∈ R n , i.e., there exists a neighbourhood U ⊆ R n of x 0 such that where (L f h)(z) := h′ (z) • f (z) denotes the Lie derivative of h along f .Then, by [15,Prop. 5.1.2],there exists a coordinate transformation Φ : where ξ 1 (t) = y(t) ∈ R m is the original output, and η denotes the internal dynamics.If γ(•) is sign definite, then, for appropriate f, T, the state-space representation ( 3) is locally equivalent to (1).In [5,27] sufficient conditions for a global transformation are formulated in terms of differential geometric properties.In the present article, structural conditions are formulated in terms of the high-gain property (Definition 2.1) and the operator class (Definition 2.2).Systems (1) are restricted to r i = r for all i = 1, . . ., m.It is possible to generalize the proposed controller to systems (y where Λ(y)(t) := (y 1 (t), ẏ1 (t), . . ., y However, since such systems massively increase complexity of notation, we restrict ourselves to systems (1).

Feedback law
We formulate the feedback law, which achieves the control objective (2).The two main ingredients are the prescribed final time T > 0, and the error boundary, i.e., the funnel function φ.To establish the controller, we introduce the following control parameters.Choose the final time T > 0, some c > 0, and the funnel function and further choose In Remark 2.5 we comment on the control parameters defined in (4).We set and recursively define for γ 0 = 0, and k = 1, . . ., r with α(•) from (4b) the functions Then, with the functions introduced in (4), (5) we define the feedback law u : R ≥0 → R m by Remark 2.5.We comment on the control parameters defined in (4), and on the high-gain property.
i) The high-gain property in Definition 2.1 is essential to achieve the control objective (2).If a large input is applied, the system has to react appropriately, i.e., if the error is close to the funnel boundary, a large input results in a "fast" response of the system.For a more detailed discussion and equivalent conditions of the high-gain property we refer to [3,Rem. 1.3 & 1.4].ii) Compared to the funnel controller proposed in, e.g., [14,3], the explicit choice of φ(•) in (4a) seems restrictive.Anticipating the initial conditions (8) in Theorem 3.1, this choice of φ(•) reflects the intuition that the shorter the final time T is chosen, the better the initial guess has to be.iii) The bijection α(•) is responsible for the high-gain, i.e., the smaller the distance between the error and the funnel boundary is, the larger the input values are.A typical choice is α(s) = c(r + 1)/(1 − s).iv) The parameter c > 0 in (4a) links the funnel function φ(•) to the gain function α(•) in (4b).The larger the value c > 0, the larger the lower bound of α(•), i.e., small tracking errors result in higher input values.From the perspective of the initial conditions (8), the parameter c > 0 can be used to satisfy these.Given a final time T > 0 and initial values, the inequalities in (8) can be utilized to find an appropriate c > 0. So in view of the initial conditions, the parameter c > 0 and the final time T have an intuitive relation: the shorter T is, the larger c must be.Moreover, the larger the initial error is, the large c must be.v) The surjection N (•) from (4c) accounts for possible unknown control directions.A feasible choice is, e.g, N (s) = s sin(s).If the control direction is known, e.g., y (r Then, with e k (•) from (5a) and γ k (•) from (5b) we obtain γ (j) k (t) = Γ j t, e k (t), . . ., e k+j (t) , for 0 ≤ j ≤ r − k, which can be seen via a induction over k using (5).
Due to the recursion (5), the controller ( 6) is not as simple to implement as the controller in [3, Eq. ( 9)].However, given (7), the calculation of the required expressions can be done completely algorithmically.

Main result
This section contains the main result.To phrase it, the application of the controller (6) to a system (1) yields a closed-loop initial value problem that has a solution; the input and output signals are bounded and in particular, the controller achieves exact output tracking in predefined finite time with prescribed behaviour of the tracking error.
Then, the funnel controller (6) applied to (1) yields an initial value problem, which has a solution and every maximal solution y : iii) the tracking error e(t) = y(t)−y ref (t) evolves within the performance funnel F φ , i.e., iv) the tracking of the reference and its derivatives is exact at t = T , i.e., The proof is relegated to the appendix.Note that, since the system class N m,r encompasses the systems under consideration in [1,21,2,11,16,29,22], and under additional regularity assumptions those in [32], the proposed feedback law (6), assuming availability of the first r − 1 output derivatives, achieves the control objectives formulated in those references with prescribed behaviour of the error and within predefined finite time.
Remark 3.2.Since at the first glance the control law ( 6) is very similar to the controller proposed in [3], we emphasize some differences.
i) As highlighted in, e.g., [13,3], some care is required when showing boundedness of the involved signals, since the bijection α(•) may introduce a singularity.Moreover, in the present context, expressions involving the unbounded funnel function φ(•) demand particularly high attention, cf.Steps two and three in the proof.ii) A careful inspection of the proof of [3,Thm. 1.9] yields, that the following growth condition [T ε , T ), this is, for all k = 0, . . ., r − 1 and all t ∈ [T ε , T ) we have ∥e (k) (t)∥ < ε.This property is relevant, e.g., if during a docking manoeuvre the demanded accuracy changes.

Numerical examples
We present two numerical examples.In the first simulation we consider a docking maneuver as an application.
The second simulation illustrates how the choice of φ affects the maximal control input.

Docking maneuver
As an exemplary application we simulate docking of two spaceships.Consider a passive space station in a circular orbit, and a chasing active spacecraft.We assume the passive space station to be on a constant altitude r s with constant angular velocity ω = µ/(r e + r s ) 3 , where µ ≈ 3.986 • 10 14 m 3 /s 2 is the standard gravitational parameter, and r e = 6378137 m the radius of the earth.To analyze the motion of the spacecraft, we use Hill's local-vertical-local-horizontal coordinate frame [12], see Figure 2. Within this frame we use the commonly used Clohessy-Wiltshire model for satellite rendezvous [7], also elaborated on in [17].Let r(t) denote the altitude of the chasing spacecraft at time t.Since lim t→T φ(t) = ∞, simulation is only possible for [0, t max ] with t max < T .Since φ(t)∥e(t)∥ < 1 for all t ∈ [0, T ), in virtue of Remark 3.4 a value eps can be chosen such that a certain upper bound of the spatial error at final time t max is guaranteed, i.e., ∥e(t max )∥ < 1 φ(tmax) = c(T − t max ) ≤ eps, from which we obtain t max ≥ T − eps/c.Here we choose eps = 10 −10 m, which means a spatial accuracy of Ångström (range of size of atoms).This seems to be a unnecessary high accuracy since in real applications the required rendezvous distance is about centimetres, then magnetic docking structures become active; however, if these fail unexpectedly, the feedback control still is capable to perform a docking maneuver.Simulations have been performed in Matlab (solver: ode15s, AbsTol=RelTol=10 −12 ). Figure 3 shows that the docking maneuver is successful within the predefined finite time T , and the errors evolve within the prescribed boundary.Figure 4 shows the errors of the velocities, where the disturbance can be seen as fast oscillations.As expected from Theorem 3.1, the errors of the velocities tend to zero for t → T .The control input is depicted in Figure 5. Figure 5b shows the control input in the very last moments before docking, where the largest input signals are generated, which may not be needed if the docking tools are activated and work as intended.

Control effort and funnel function
To illustrate how the choice of φ in (4a) influences the control input, we consider a chain of integrators ÿ(t) = u(t), y(0) = 1, ẏ(0) = 0,   Second, the funnel boundary is 1/φ(t) = c(T − t).So larger values of c result in a faster decay of the boundary, and so the distance of the error to the funnel boundary decreases faster.Hence, larger input values are required to push the error away from the boundary.

Conclusion
We proposed a feedback controller, which achieves exact tracking in predefined finite time, while the tracking error evolves within prescribed boundaries.We rigorously proved boundedness of all signals, and that the error as well as all its relevant derivatives vanish at the predefined final time.
ful for the comments of the anonymous reviewers who drew my attention to some important aspects.
Appendix: Proof of Theorem 3.1 First, we state the followin result.
Proof of Theorem 3.1.The proof consists of eight steps.
Step six.We show ω = T .Via the previous steps we have for all k = 1, . .Step eight.We show that the tracking error e(•) and its derivatives tend to zero as t → T , this is, we show ∀ k = 1, . . ., r : lim t→T ∥e (k−1) (t)∥ = 0.
) has a solution in the sense of Carathéodory, meaning a Nomenclature.[a, b], [a, b), (a, b) is a closed, half-open, and open interval for a, b ∈ R, a < b; R ≥0

Remark 3 . 3 .Remark 3 . 4 .
for almost all t ≥ 0 on the funnel function ϕ is crucial.It prevents a "blow up" in finite time, i.e., ϕ(•) is bounded on any compact interval.With this, however, exact tracking in finite time via funnel control is impossible.Contrary, the funnel functions φ(•) in (4a) do not satisfy this growth condition.Hence, the respective steps in the proof of[3, Thm.1.9]  are not valid in the present analysis.iii) In order to show boundedness of the involved error signals, novel techniques have been developed in the proof of Theorem 3.1.In particular Steps two, three and four contain innovations not found in the existing works on high-gain feedback control.iv) The conclusion drawn in Step eight, namely that the tracking error is zero at t = T , is only possible with the results derived in Step three.Assertion i) in Theorem 3.1, namely [0, T ) being the maximal solution interval, naturally raises the question of a global solution in time.If the system's equations (1) are available and y ref (•) is defined on R ≥0 , then y(t) − y ref (t) ≡ 0 for all time t ≥ T can be achieved by asking the reference to satisfy (1) for t ≥ T , with u(•) ≡ 0, and "initial" conditions y ref (T ) = y(T ), ẏref (T ) = ẏ(T ), . . ., y (r−1) ref (T ) = y (r−1) (T ).For any given ε > 0 there exists a time T ε < T such that each of the first r − 1 derivatives of the error e(•) = y(•) − y ref (•) can be bounded by ε for all t ∈
Control input in the very last moments.

Figure 6 .
Figure 6.Maximal input for different values of c in (4a).andperform stabilization, i.e., y ref (t) ≡ 0, with exact value at final time T = 1.To influence the shape of φ, the parameter c is varied with c k = 2 + 0.1k for k = 0, . . ., 200, while keeping the final time T constant.Figure6shows the relation between c and the maximal control input ∥u∥ ∞ .The maximal applied control increases with increasing c.This can be understood via the following reasoning.First, in virtue of Remark 2.5 iv), larger values of c cause larger lower bounds of the bijection α(•).Second, the funnel boundary is 1/φ(t) = c(T − t).So larger values of c result in a faster decay of the boundary, and so the distance of the error to the funnel boundary decreases faster.Hence, larger input values are required to push the error away from the boundary.