8.6 Nonlinear Observer Design
8.6.2 Nonlinear Local Observer
The observer presented in the previous section prominently lends itself to the imple-mentation of an observer for the variable displacement vane pump system. However, exploring other observer concepts is worthwile with regard to practical questions such as the measurability of the system output y = x1. As an alternative to the high gain observer presented in the preceding section, a nonlinear local tracking observer based on linearization of the system dynamics about a desired or nominal trajectory in state space x∗ can be constructed. The idea behind the corresponding observer is to choose the observer gain l(x) such that the linearized error dynamics are asymptotically stable.
The observer gain is chosen according to a pole-placement technique for time-varying systems. The concept is extensively discussed in [87, 88, 89, 132] in the context of differ-entially flat systems to which it can be applied in the most straightforward fashion. In physical state coordinates x, the observer has the representation
˙ˆx = fd(ˆx) + g(ˆx)ˆu + l(t)(y− h(ˆx)) , (8.205) where in contrast to the high gain form observer presented before y = h(ˆx) does not need to be cam ring displacement but can be any other state variable in agreement with the observability criterion.
Combining (8.205) with the system dynamics given by (7.1), (7.2) yields dynamics for the observation error ˜x = ˆx − x which are of the following form:
˙˜x = f (ˆx) + g(ˆx)ˆu− f (x)− g(x)ˆu + l(t) h(x) − h(ˆx)
, (8.206)
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 8.6: 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 8.7: With valve dynamics.
˜
x(0) = ˆx0− x0. (8.207)
It is to be emphasized that the above representation of the observer error dynamics expresses the fact that the system input u = ˆu is synthesized as a function of the state estimates provided by the observer.
Now linearizing about x∗ = x∗(t), u∗ = u∗(t)yields
Here, the time varying matrices are
A(t) = ∂(fd(x)+g(x)ˆu) with v = v(t) and A = A(t), the time-varying observability map for a system with n states can be defined as follows:
O(cT(t), A(t)) =
is called observable if rank O(cT(t), A(t))
= n. For the SISO system considered at this stage, this is equivalent to showing the determinant of the observability matrix does not become zero along the desired trajectory x∗, u∗. It can easily be seen that applying the definition for the observability of a time-varying system to a time-invariant system yields the observability matrix well known from linear time invariant theory. If observ-ability is given, the system can be transformed to so called observobserv-ability normal form.
To do so, another operator first requires introduction. The operator
N0 ◦ ( · ) = ( · ) , (8.215)
NA1 ◦ ( · ) = −dtd (· ) + A ( · ) , (8.216) NAi ◦ ( · ) = NA1
NAi−1◦ ( · )
(8.217) allows for the construction of the inverse of matrix transforming the system to observ-ability normal form.
The inverse of the transformation matrix is computed as V−1(t) =
and ˜cn−1(t) constituting a degree of freedom in the design of the observer. It ought to be chosen in such a way that v(t) assumes the simplemost form. Typically, an effort will be made to choose ˜cn−1(t)in such a way that it will render the representation for v(t) constant, if possible. Applying the transformation z(t) = V(t)∆x(t) yields the observability normal form
Transforming the linearized error dynamics to observability normal form with ζ(t) = V(t)∆˜x(t), one gets
The idea then is to chose ˜l(t) in a way that renders the matrix E(t) time invariant and assigns poles with strictly negative real parts to E(t) = E. Transforming back to ∆˜x coordinates, the observer gain required for asymptotic stabilization of the observer error is l(t) = V−1(t)˜l(t).
In the case of the present variable displacement vane pump system, one obtains
A(t) =
The control input matrix for the linearized system is
B(t) =
BL(t) =
0 0
1
Ch1sign(x∗3− x∗4)γFbReg
p|x∗3− x∗4|
−C1hSsign(x∗3− x∗4)γFbReg
p|x∗3− x∗4|
(8.229)
and
cT = [0 0 1 0] . (8.230)
To evaluate the above matrices, the nominal trajectory x∗ which is being linearized about has to be generated. As a differentially flat output is not available for the system, the generation of a nominal trajectory requires the integration of the internal dynamics as excited by the external dynamics ξ∗. This integration is complicated by the switching behavior of the internal dynamics as a function of valve input uR. Given a desired tra-jectory x∗ ∈ R4, however, the corresponding nominal control input u∗Rcan be computed as well so that linearization of the system dynamics about x∗, u∗R can be performed.
It is to be remarked at this point that the approach outlined above and applied to the system with switched control input matrix is expected to yield asymptotically stable error dynamics as the eigenvalues of the error dynamics are identically placed for both tank-sided and load-sided operating conditions. While this is intuitive, the observer gain calculation depends on each system configuration’s normal form. These normal forms are different for tank-side and load-side flow condition, as AT 6= AL. Therefore, z(t) will not have the same meaning when applied to the tank-side and the load-side flow condition, respectively, so that in fact one obtains zT(t) and zL(t), respectively. Ultimately, each system configuration will yield asymptotically stable error dynamics as derived from the corresponding normal form – this will, however, not generally guarantee asymptotic stability of the switched error system in a strict sense as is easily demonstrated by the example from Branicky presented in the intro-duction as stability will also depend on the switching law that switches between the tank-side and load-side flow condition. Because the switching behavior up to this stage of analysis has not yet shown any dysfunctional properties, it is likely that the observer approach based on system linearization will work and should therefore be investigated.
In the next two subsections, two nonlinear local observers based on system linearization about the desired trajectory are thus presented. Observer I makes use of two separate pressure measurements, p1 and pS, or x3 and x4 in state space coordinates. Observer II is constructed as a full observer and requires measurement of only p1 = x3 for the estimation of all states.
8.6.2.1 Nonlinear Local Observer I
Assuming the availability of an additional measurement of x4, it is possible to construct a reduced observer for which only an estimation of x1, x2 and x3 needs then to be per-formed as x4 is taken to be known from measurement – it is the very knowledge of x4
that allows for the construction of this reduced observer only partially estimating states.
The matrices (8.224) and (8.225) can thereby be reduced by the fourth column and row each, allowing for a simplified observer design. This additional measurement may not only improve the observer performance in general, but can possibly be relevant in case the capacitance ChScan not be taken constant. Because the associated volume is smaller than that of the main capacitance, ChS is much more prone to showing a dependence on pump displacement than is Ch1. With an appropriate choice of the design parameter
˜
cn−1(t), the simplemost structure of v(t) is the same for both flow conditions:
v(t) = vT(t) = vL(t) = ex-plicitly – in its derivation, the symbolic computation abilities of MAPLE were intensely used. From this vector, the repeated application of the N-operator allows the deriva-tion of the three-component error-stabilizing observer gain also known as time-variant Ackermann formula
where the ai, i = 0, 1, 2are the coefficients of the characteristic polynomial defining the desired pole locations for the observer.
8.6.2.2 Nonlinear Local Observer II
If only a measurement of p1 = x3can be made and the displacement dependence of ChS
is negligible for the linearization quality about x∗, u∗R, a full observer can be constructed in similar fashion as described above for a reduced observer.
Here, the respective operators are applied to the full linearization matrices from (8.224) and (8.225), starting from a vector v(t) for the tank side which, through an appropriate choice of ˜cn−1(t)assumes the form
The structural similarity of this vector in its first three components with (8.231) is evi-dent. As for the corresponding vector vL(t), its length is prohibitive for stating it explic-itly. The resulting observer gains for each flow condition, however, are
lT(t) = 1
˜
cn−1,T(t) h
a0NA0TvT(t) + a1NA1TvT(t)
+a2NA2TvT(t) + a3NA3TvT(t) + NA4TvT(t)i
, (8.235)
lL(t) = 1
˜
cn−1,L(t) h
a0NA0LvL(t) + a1NA1LvL(t)
+a2NA2LvL(t) + a3NA3LvL(t) + NA4LvL(t)i
(8.236) with ai, i = 0, . . . , 3again being the coefficients of the desired characteristic polynomial for the observer.
8.6.2.3 Simulation Results
Simulation results for the reduced observer are shown in Figure 8.8.
The results indicate that in principle, state estimation with the reduced observer is pos-sible. A closer look at the results in Figure 8.9 reveal, however, that the convergence behavior of the observer featuring valve dynamics is not ideal as a comparatively ex-tensive transient is evident in the simulation results. While convergence was achieved in the example here, other trajectories to be tracked yielded non-converging results. In addition, extensive simulations revealed that the observer is sensitive with respect to observer initial conditions and numerically sensitive, too.
For the full observer, simulation results are shown in Figures 8.10 and 8.11.
While observer and control convergence was achieved, it should be pointed out that this rests heavily upon a significantly increased mass mP S for these simulations: instead of 0.32kgas in all other simulations, a total mass of 1kg had to be chosen in order to achieve the results shown in Figure 8.10. This makes the applicability of the full nonlinear local tracking observer questionable. As for both reduced and full nonlinear local tracking observer, numerical instability and sensitivity towards observer initial conditions are an issue frequently encountered during simulation, see Figure 8.12 for an example of a non-converging solution.
Ultimately, this observer type is far from being implemented easily. To conclude, even though theoretically applicable in some cases, the nonlinear local tracking observers can not be expected to generally yield satisfactory results in the context of this system so that only state estimation based on cam ring displacement appears to be a feasible observer approach.
0 0.5 1 1.5
(b) Main chamber pressure.
0 0.5 1 1.5
(c) Actuation chamber pressure.
Figure 8.8: Reduced observer: Without valve dynamics.
(b) Main chamber pressure.
0 0.5 1 1.5
(c) Actuation chamber pressure.
Figure 8.9: Reduced observer: With valve dynamics.
0 0.5 1 1.5
(b) Main chamber pressure.
0 0.5 1 1.5
(c) Actuation chamber pressure.
Figure 8.10: Full observer: Without valve dynamics.
(b) Main chamber pressure.
0 0.5 1 1.5
(c) Actuation chamber pressure.
Figure 8.11: Full observer: With valve dy-namics.
0 0.5 1 1.5 0
1 2 3
·10−3
t [s]
xS,ˆxS,x
∗ S
[m]
x∗S ˆ xS xS
Figure 8.12: Pump displacement for non-convergent simulation – non-zero consumer pressure pC > 0.