The objective is that the outputy of the system (5.3) tracks a given reference signal which isyref∈W1,∞([0,∞);Rm) with a prescribed performance of the tracking error e:=y−yref, that iseevolves within the performance funnel
Fϕ:={(t, e)∈[0,∞)×Rm |ϕ(t)kekRm<1 }
5.3. FUNNEL CONTROL 135
defined by a functionϕbelonging to Φγ :=
ϕ∈W1,∞([0,∞);R)
ϕ|[0,γ]≡0, ∀δ >0 : inf
t>γ+δϕ(t)>0
, for some γ >0. The situation is illustrated in Fig. 5.1. The funnel boundary given by 1/ϕis unbounded in a small interval [0, γ] to allow for an arbitrary initial tracking error. Since ϕ is bounded there exists λ > 0 such that 1/ϕ(t) ≥ λ for all t > 0.
Thus, we seek practical tracking with arbitrary small accuracyλ >0, but asymptotic tracking is not required in general.
t
±ϕ(t)−1 γ
±λ−1 e(t)
Figure 5.1: Error evolution in a funnel Fϕ with boundaryϕ(t)−1.
The funnel boundary is not necessarily monotonically decreasing, while in most situ-ations it is convenient to choose a monotone funnel. Sometimes, widening the funnel over some later time interval might be beneficial, for instance in the presence of peri-odic disturbances or strongly varying reference signals. For typical choices of funnel boundaries see e.g. [56, Section 3.2].
A controller which achieves the above described control objective is the funnel control-ler. In the present chapter it suffices to use the simple version developed in [61], which is the feedback law
Is,e(t) =− k0
1−ϕ(t)2kB0v(t)−yref(t)k2Rm
(B0v(t)−yref(t)), (5.6) wherek0>0 is some constant used for scaling and agreement of physical units. Note that fort≤γ, the controller is merely
Is,e(t) =−k0(B0v(t)−yref(t)), sinceϕ|[0,γ]≡0.
136 PROBLEM The application of the controller (5.6) results in the nonlinear and time-varying PDE system
We are interested in considering solutions of (5.7), which leads to the following weak solution framework.
Definition 5.3.1. Assume thatd≤3 and Ω⊂Rdbe a bounded domain with Lipschitz boundary, let D ∈ L∞(Ω;Rd×d) be symmetric-valued and satisfying the ellipticity condition (5.1). LetT ∈(0,∞] andAbe the Neumann elliptic operator on Ω associated toD(see Proposition 5.1.1), letB ∈L(Rm,(W1,2(Ω))0), and letu0, v0∈L2(Ω) as well as Is,i ∈L2loc([0,∞);L2(Ω)) be given. Further, letk0 >0, yref ∈W1,∞([0,∞);Rm), γ >0 andϕ∈Φγ. A triple of functions (u, v, y) is calledsolution of the system (5.3) with feedback (5.7) in [0, T), if (u, v, y) is a solution of (5.3) in [0, T) with (5.6) for all t∈[0, T).
Remark 5.3.2.
a) Plugging the feedback law (5.6) into the system (5.3), we obtain d
dtv(t) =Av(t) +p3(v)(t)−u(t) +Is,i(t)− k0B(B0v(t)−yref(t)) 1−ϕ(t)2kB0v(t)−yref(t)k2Rm
, d
dtu(t) =c5v(t)−c4u(t),
(5.7) Consequently, (u, v, y) is a solution of system (5.3) with feedback (5.7), if, and only if
(i) v∈L2([0, T);W1,2(Ω))∩C([0, T);L2(Ω))) withv(0) =v0; (ii) u∈C([0, T);L2(Ω)) withu(0) =u0;
(iii) for all χ ∈ L2(Ω), θ ∈ W1,2(Ω), the scalar functions t 7→ hu(t), χi, t 7→
hv(t), θiare weakly differentiable on [0, T], and it holds that, for almost all t∈(0, T),
d
dthv(t), θi=−a(v(t), θ) +hp3(v(t))−u(t) +Is,i(t), θi
− k0hB0v(t)−yref(t),B0θiRm
1−ϕ(t)2kB0v(t)−yref(t)k2Rm
,
d
dthu(t), χi=hc5v(t)−c4u(t), χi, y(t) =B0v(t),
(5.8)
The system is a nonlinear and non-autonomous PDE and any solution needs to satisfy that the tracking error evolves in the prescribed performance funnel Fϕ. Therefore, existence and uniqueness of solutions is a nontrivial problem and even if a solution exists on a finite time interval [0, T), it is not clear that it can be extended to a global solution.
5.3. FUNNEL CONTROL 137 b) For global solutions it is desirable thatIs,e∈L∞([δ,∞);Rm) for allδ >0. Note
that this is equivalent to lim sup
t→∞
ϕ(t)2kB0v(t)−yref(t)k2Rm <1.
It is as well desirable thatyandIs,e have a certain smoothness.
In the following we state the main theorem of this chapter. We will show that the closed-loop system (5.7) has a unique global solution so that all signals remain bounded.
Furthermore, the tracking error stays uniformly away from the funnel boundary. We further show that we gain more regularity, ifB ∈L(Rm,(Wr,2(Ω))0) for somer∈[0,1).
Theorem 5.3.3. Assume that d≤3 andΩ⊂Rd is a bounded domain with Lipschitz boundary, letD∈L∞(Ω;Rd×d)be symmetric-valued and satisfying the ellipticity con-dition(5.1). LetAbe the Neumann elliptic operator onΩassociated toD(see Proposi-tion 5.1.1), letB ∈L(Rm,(W1,2(Ω))0)withkerB={0}, and letu0, v0∈L2(Ω)as well as Is,i ∈ L∞([0,∞);L2(Ω)) be given. Further, let k0 >0, yref ∈ W1,∞([0,∞);Rm), γ > 0 and ϕ ∈ Φγ. Then, for all T > 0, there exists a unique solution of (5.7) in [0, T). The solution is thus global and
(i) u,u, v˙ ∈BC([0,∞);L2(Ω));
(ii) for all δ >0holds
v∈C0,1/2([δ,∞);L2(Ω)), y, Is,e∈BU C([δ,∞);Rm), and
(iii) there exists some ε0>0 such that for allδ >0
∀t≥δ:ϕ(t)2kB0v(t)−yref(t)k2Rm ≤1−ε0.
a) If, further,B ∈L(Rm,(Wr,2(Ω))0)for somer∈(0,1), then for allδ >0 v∈C0,1−r/2([δ,∞);L2(Ω)),
y, Is,e∈C0,1−r([δ,∞);Rm),
b) If B ∈L(Rm, L2(Ω)), then for all δ >0and for all λ∈(0,1) v∈C0,λ([δ,∞);L2(Ω)), y, Is,e∈C0,λ([δ,∞);Rm),
c) If B ∈L(Rm, W1,2(Ω)), then for all δ >0 holdsy, Is,e∈C0,1([δ,∞);Rm).
138 PROBLEM If in particular,v0∈ D(A), then the former hold withδ= 0.
Remark 5.3.4.
a) The condition kerB={0} is equivalent to imB0 being dense inRm. The latter is equivalent to imB0=Rmby the finite-dimensionality of Rm.
Note that surjectivity of B0 is evident for tracking control, since any reference signal yref∈W1,∞([0,∞);Rm) has to be tracked by the outputy=B0v.
b) If the input operator corresponds to Neumann boundary control, i.e. B is as in (5.5) for some w ∈ L2(∂Ω), then r ∈ (1/2,1), see Remark 5.2.2d). Con-sequently, for all ε > 0, we have v ∈ C0,3/4−ε([δ,∞);L2(Ω)) and y, Is,e ∈ C0,1/2−ε([δ,∞);Rm) in this case.
c) If the input operator corresponds to distributed control, i.e. Bu=uw for some w∈L2(Ω), see Remark 5.2.2c), then for allε >0 holdsv∈C0,1−ε([δ,∞);L2(Ω)) andy, Is,e∈C0,1−ε([δ,∞);Rm).
Before we begin to develop the necessary results to prove Theorem 5.3.3, we show the simulated system (5.7).
A numerical example
In this section, we illustrate the practical applicability of the funnel controller for a numerical example. The setup chosen here is a standard test example for termina-tion of reentry waves and has been considered similarly in, e.g., [20, 80]. All simu-lations are generated on an AMD Ryzen 7 1800X @ 3.68 GHz x 16, 64 GB RAM, MATLABR Version 9.2.0.538062 (R2017a). The solutions of the ODE systems are ob-tained by the MATLABR routineode23. The parameters for the FitzHugh-Nagumo model (5.3) used here are as follows:
Ω = (0,1)2, D=
0.015 0 0 0.015
,
c1
c2 c3 c4
c5
≈
1.614 0.1403
0.012 0.00015
0.015
.
The spatially discrete system of ODEs corresponds to a finite element discretization with piecewise linear finite elements on a uniform 64×64 mesh. The uncontrolled sys-tem is stimulated such that reentry phenomena occur, see Figure 5.2. Let us emphasize that the associated dynamics are obtained from (5.3) withIs,i= 0 =Is,e.
For the control interaction, we assume thatB ∈L(R4,(W1,2(Ω))0) where the Neumann control operator is defined such that
B∗z= R
Γ1z(ξ) dσ,R
Γ2z(ξ) dσ,R
Γ3z(ξ) dσ,R
Γ4z(ξ) dσ>
,
5.3. FUNNEL CONTROL 139
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Figure 5.2: Snapshots of reentry waves fort= 100 (left) andt= 200 (right).
Γ1={1} ×[0,1], Γ2= [0,1]× {1}, Γ3={0} ×[0,1], Γ = [0,1]× {0}.
Note that this scenario is the one described in Remark 5.2.2 d), so Theorem 5.3.3 and Remark 5.3.4 b) imply that the solutiony, Is,e∈C0,1/2−ε([δ,∞);Rm) for allε >0.
The functionϕcharacterizing the performance funnel (see Figure 5.3) is chosen as ϕ(t) =
(0, t∈[0,0.05], tanh(100t ), t >0.05.
0 50 100 150 200 250 300 350 400 450 500
0 1 2 3
t
1 ϕ(t)
ke(t)kRm
Figure 5.3: Error dynamics and funnel boundary.
The simulations correspond to a finite element discretization of (5.3) with piecewise linear finite elements.
For replication of a natural (desired) heart rhythm, the funnel reference signal is ob-tained by a periodic excitation wave spreading out from the stimulation point in the center of the domain. The smoothness of the signal is guaranteed by convoluting the
140 PROBLEM original signal with a triangular function. Figure 5.4 show the results of the closed-loop system obtained with the control law
Is,e(t) =− 0.75
1−ϕ(t)2kB0v(t)−yref(t)k2Rm
(B0v(t)−yref(t)),
which is visualized in Figure 5.5. The initial condition is taken as a snapshot of the reentry wave which without control will not terminate. Let us note that the sudden changes in the feedback law are due to the jump discontinuities of the intracellular stimulation currentIs,i used for simulating a regular heart beat.
0 200 400
0 50 100
t
y
y1,ref
y1
0 200 400
0 50 100
t
y
y2,ref
y2
0 200 400
0 50 100
t
y
y3,ref
y3
0 200 400
0 50 100
t
y
y4,ref
y4
Figure 5.4: Reference signals and outputs of the funnel controlled system.
It is seen from Figure 5.4 that the controlled system faithfully reproduces the desired reference signal except for the very beginning where the control is not active yet. Figure 5.5 further shows that this is achieved with a comparably small control effort.