3. Novel perspective in the Control of Nonlinear Systems via a Linear Param-
3.5. Observer based control for LPV systems
In this section, I will investigate a completed observer structure, which is based on the same consideration as the previous Sections. This method uses the classical state observer theorem, however, the completed observers are parameter dependent and fit to the developed completed controller structure.
I demonstrate the usability of the completed observer structure on the model of Sections 3.4.1.
3.5.1. Classical linear observer design
It may happen that the states of a given system are directly not available. In such situations, a state observer or state estimator can be used. Although other possibilities are available, e.g. family of Kalman filters, I only focused on the classical observer design.
A linear observer can be designed, if the main criteria, the observability is satisfied.
That means, that the rank of the observability matrix has to be equal ton, the number of the states. This property refers to the structure of theAandCmatrices. If observability is satisfied, linear observer design is possible.
In case of full order linear observer, the linear observer is a dynamic system whose output is the ˆx(t) estimated state vector. In case of asymptotic observer, the estimation error, namely ˜x(t) :=x(t)−x(t) has to converge to zero over time .ˆ
The general form of the full order linear observer can be described as follows:
x(t) =Fˆx(t) +Gy(t) +Hu(t) , (3.47) where F∈Rn×n is the observer state matrix, G∈Rk×n is the observer gain matrix and H∈Rn×m is the observer input matrix.
The velocity of the disappearance of the observation error can be prescribed by the eigenvalues of the F, which is traceable to the determination of the characteristic polynomial ofF[56,88]:
|sI−F|=|sI−A−GC|=|sI−A>−C>G>| . (3.48) In this way an asymptotic state observer design leads to a state feedback design task, where the G observer gain provide that the closed loop poles of the observer become equal to the predefined observer poles.
In order to design a full order observer, the following criteria have to be satisfied by
which the observer’s parameter can be calculated at the same time [56,88]:
x(t) =F˜x stable and fast
The selected observer poles have to be higher negative values than the LTI system has in order to provide fast operation, namely, fast error disappearance.
3.5.2. Completed parameter dependent observer design
In this case, I demonstrate that Theorems 3.3.1and 3.3.2can be used in the same way as in (3.19) and (3.20) to design state observer, based on similar principles.
Let Fref = Aref −GrefC∼ F(t) = A(p(t))−(Gref +G(t))C, which means that the eigenvalues of Fref, the λ(Fref) and λ(F(t)) become equal during operation. So, λ(Fref) =λ(F(t)) at ∀p(t), if λ(F(t)) are the eigenvalues ofF(t) =A(p(t))−(Gref + G(t))C. This is only possible if the similarity transformation matrix is theIn×n unity matrix, namely,Fref =I−1F(t)I. As previously, that means that the introduced observer gain has to provide the ”smoother” similarity, but also the ”strict” equality criteria as well.
Shortly, the proposed completed observer gainGref +G(t) has to provide the equality of not just the eigenvalues λ(Fref) =λ(F(t)), but also the equality of the matrices, as well:
3.5.3. Consequences, observer design and limitations
Consider the case when p(t) can be measured or estimated. In this case, G(t) can be calculated via rearranging the (3.50) at everyp(t):
(Aref−GrefC−A(p(t)) +GrefC)C−1=−G(t)CC−1 G(t) =−(Aref−GrefC−A(p(t)) +GrefC)C−1 G(t) =−(Aref−A(p(t)))C−1
By rearranging (3.51):
Gref −(Aref −A(p(t)))C−1
an observer structure appears, which can ensure that the parameter dependent observer is going to behave as the reference observer itself, regardless of the actual value ofp(t).
Similar to the previous Section, I have assumed that the parameter vector p(t) can be measured or estimated, which is an important limitation. However, most of the cases if the parts of p(t) cannot be measured, it can be estimated by the observer itself, or external estimator also can be used.
The control structure with the completed observer can be seen on Fig. 3.17.
N(p(t)) Controller Kref + K(t)
LPV system S(p(t))
F(t)x(t) + (Gref + G(t))y(t)+ Hu(t) u(t)
Figure 3.17.: General feedback control loop with completed observer and gain The key points which are needed in order to realize the proposed completed observer design method can be summarized as follows – which should follow the afore mentioned realization steps of the completed controller:
1. Realization of the completed LPV controller structure;
2. State observer design via linear controller design methods in order to realize the observer gainGref for the reference LTI system S(pref);
3. Design of the eligible observer scheme, including the appropriate form of (3.51);
4. Realization of the control environment.
I present the main limitations and their possible solutions below.
1. The first step is the same as the completed controller design.
2. The invertibility of the output matix C is the key point. The structure of a system’s output and the output equation is structurally different than the input.
That means, we have to face strict limitations. Two bottleneck has to be investigated, the mathematical and the control one. Mathematically, theCmatrix has to be a square matrix and has to be invertible. The control interpretation of this restriction is that the number of the output (and thus the number of the output equations) has to be equal to the number of the states, namely, each state has to be an output, or should has direct affect on the output. This prescription makes the columns of theCmatrix linearly independent and thus invertible.
There is a chance that this property can be handled through mathematical transfor-mations, which transforms the completed equation of (3.51) into an other abstract space (by using basis transformations), however, I did not investigated this possi-bility yet.
In the following section, I demonstrate how the completed controller and observer design works in practice on given nonlinear physiological system.