Boundedness of Tracking

Applying the control strategy presented in this chapter separates the state variables x into observable transformed state variables ξ and unobservable state variables η ={ηT, ηL}. The latter correspond to internal dynamics and zero dynamics associated with these internal dynamics under the assumptions of the output-zeroeing problem.

While for the external dynamics asymptotic tracking is achieved by means of nonlin-earity compensation and appropriate pole-placement, the boundedness of the full state xis a requirement for the control strategy to be feasible. Since the state x is obtained from the internal variables ξ and η through the inverse of Φ(x), once the internal variables lack stability, they possibly become unbounded, thereby rendering the state xunbounded. This evidently is a problem because from a certain point onwards, the nonlinearity compensation will become impossible due to physical bounds on the input required to compensate nonlinearities featuring unbounded states. It therefore remains to be shown that the system yields bounded states for the two possible configurations of the control input matrix gR.

In this section, it will be shown that the suggested control approach leads to the desired outcome. The approach relies on a theorem suggested in [40]. Because for input-output linearization (i.e. systems with non-full relative degree) it is not asymptotic stability for the tracking error of all transformed states but only boundedness for the full state vector that is required, it follows that if both control systems (tank-sided and load-sided valve opening, i.e. uR ≥ 0 and uR< 0) can be shown to yield bounded tracking, then the sys-tem will show bounded tracking as a whole – the non-smoothness of the control input matrix hence is of no relevance for the functionality of the proposed control approach.

In [91], an alternative theorem on the boundedness of tracking is proposed that is fre-quently drawn onto in the extant body of research. It does, however, require global

exponential stability of the zero dynamics – a condition that is not fulfilled here due to only locally exponentially stable zero dynamics. For such cases, theorem 1 due to Isidori [40] (see Appendix A) states sufficient conditions for bounded tracking (see also [97]) .

According to this theorem, in order to show that the tracking is bounded in ξ, η, it has to be shown that η(t) is bounded and uniformly asymptotically stable upon excitation through ξR for both flow conditions. The assumption here is that for the external co-ordinates ξ, exact tracking is achieved so that ξ = ξR drive the internal dynamics. To show boundedness of the response of ηT and ηLwith respect to their excitations by the reference signals theorem 3 from [53] is drawn onto (see Appendix A).

The first part of theorem 3 makes a statement on the conditions of boundedness while the second part provides correlating existence conditions that may in some cases allow calculating the bounds.

To show boundedness of ηT = qT(ξR, ηT), the Lyapunov candidate

V = 1

2Ch1η_T² (8.70)

is assumed which is radially unbounded and positive definite. It thereby is guaranteed that α1 and α2according to theorem 3 exist.

With ξR = [ξ1R ξ2R ξ3R]^T, the time derivative of V will be then manipulated as fol-lows:

V = η˙ T (ˆqPξ1R+ AS2ξ2R)− |η^T|p

|η^T|γ^FAOR1 (8.71)

≤ |ηT| q

ξ_1R² + ξ²_2R q

(ˆq_P)²+ (A_S2)²

| {z }

=: fT(t)

−|ηT|p

|ηT|γFA_OR1 (8.72)

=|η^T|f^T(t)− (1 − θ)|η^T|p

|η^T|γ^FAOR1− θ|η^T|p

|η^T|γ^FAOR1 (8.73)

≤ |ηT|KfT − (1 − θ)|ηT|p

|ηT|γFAOR1− θ|ηT|p

|ηT|γFAOR1 (8.74) with KfT representing the upper bound on fT resulting from bounded ξ1Rand ξ2R and 0≤ θ ≤ 1. Hence,

V˙ ≤ −(1 − θ)|ηT|p

|ηT|γFAOR1, ∀p

|ηT| ≥ K_fT θ

1 γFAOR1

. (8.75)

Thus, for bounded reference signals ξ1R and ξ2Rit follows that ηT is bounded.

The same approach can be used to show boundedness of ηLsubject to excitation through the reference signal ξR.

With (8.58) and (8.63) , the dynamics of ηLhave the representation

˙η_L =

introducing a new coordinate ζ is sensible:

and choosing the Lyapunov function candidate V = 1

2ζ². (8.80)

With (8.66), the Lyapunov function’s time derivative then is

V = ζ ˙ζ˙ (8.81) respectively and LΨ_3ηL is the lower bound on Ψ3ηL. The necessity of these functions to be bounded may impose restrictions on the choice of ξR. Ultimately, it follows that

V˙ ≤ −(1 − θ) 1

so that solutions for ζ are bounded by the boundedness of the reference trajectories ξR, Ψ˙4, f(t) and Ψ3ηL. Now, because ζ is bounded, so is ηL.

In order to show uniform asymptotic stability, the dynamics of the difference between the actual trajectory ηT and the reference trajectory ηR = ηT Rfor the tank-sided internal dynamics ηT arising from perfect tracking of ξR

∆ηT = ηT − ηT R (8.89)

is considered locally through linearization about ηT R. The implication then is that dis-turbances imposed on the initial conditions in ηT or along the tracked trajectory are restricted in their magnitude in a way that a linearized system description remains fea-sible. For the tank-sided operating condition, the error dynamics from (8.89) read

∆ ˙η_T =−γFAOR1

Ch1

1 2p

|η^{T R}|∆η_T , (8.90)

which are uniformly asymptotically stable for all times t ≥ t⁰ according to theorem 2 (see Appendix A) by the boundedness for ηT (and ηT Ras well) as shown above: with a Lyapunov candidate as in equation (8.70), it holds that

V =˙ −γFAOR1

1 2p

|ηT R|∆η_T² ≤ −γFAOR1

1 2p

|K^ηT R|∆η_T² , (8.91) where KηT R is the bound on the tank-sided reference trajectory for ηT so that the in-ternal dynamics are uniformly asymptotically stable. Considering the load-sided op-erating condition, the same approach can be applied accordingly, yielding the same result so that bounded tracking with uniformly asymptotically stable internal dynam-ics is ensured for both operating conditions from which bounded tracking follows for the system as a whole. Thus, because solutions for ηR are bounded and at least locally uniformly asymptotically stable, the control law devised by equations (8.36), (8.37) and (8.62) yields asymptotic output tracking with bounded states irrespective of the operat-ing condition. Ultimately, the system as a whole can stably be controlled for trajectory tracking.

8.4.4 Simulation Results

The simulations results in Figures 8.3 and 8.4 clearly show a high tracking performance for the pump eccentricity and thereby for volume flow control. Overshoots are marginal and subject to linear feedback design with valve dynamics showing a small effect on tracking behavior mainly in a transient phase. The simulation results thereby clearly indicate positive behavior of the input-output linearization.

0 0.1 0.2 0.3 0.4 0.5

(b) Main chamber pressure.

0 0.1 0.2 0.3 0.4 0.5

(c) Actuation chamber pressure.

Figure 8.3: Without valve dynamics.

0 0.1 0.2 0.3 0.4 0.5

(b) Main chamber pressure.

0 0.1 0.2 0.3 0.4 0.5

(c) Actuation chamber pressure.

Figure 8.4: With valve dynamics.