3. Novel perspective in the Control of Nonlinear Systems via a Linear Param-
3.6. Illustrative example for the completed observer structure
3.6.1. Control of nonlinear compartmental model beside observer
In this example I demonstrate the developed completed observer solution on the same system as Subsection 3.4.2. The only difference is the modification of the outputs in order to satisfy the requirement against the invertibility ofC.
Consider the compartmental system of (3.26) with modified y1,2(t) outputs:
The modified output matrix Cbecomes different than (3.27):
C=
The selected scheduling variables were p(t) =
which means we have a 3Dparameter space. As previously, these ”state based” scheduling variables are not directly measurable and in the given case they will provided by the observer.
Assume that the reference parameter vector is pref = [0.6667,0.6,0.64]>.
I have applied the same reference controller Kref for the same reference LTI system S(pref) as in Subsection 3.4.1.
The next step was the inspection of observability. The rank of the observability matrix was 2, meaning that the reference LTI system was observable.
I have designed the reference observer gain Gref by using the MATLABTM place command [99]. The obtained Gref was the following:
Gref =
Afterwards, the realization of the completed observer structure is possible (as in Fig.
3.17).
I have used the same protocol during my investigations: I compared the controlled LPV system (without observer) to the controlled and observed LPV system.
In this case, I applied a lower bound saturation on the control input signal, namely, the control signal cannot be lower than zero in both cases.
The results can be seen on Fig. 3.18. The upper left figure shows the varying of the state variables of the controlled LPV system, while the top right figure shows the changing of the estimated state variables provided by the completed observer. The lower left diagram shows the outputs of the controlled LPV system, while the lower right diagram represents the output of the controlled and observed LPV system. It can be seen that both systems reach their desired steady state values without static error and there is no difference between the outputs and states of the given LPV systems. However, as Fig. 3.19. shows, a small, oscillating error occurred between the states and outputs of the systems. The order of the error is around 10−3 – 10−4.
Time [min]
0 1 2 3 4 5 6 7 8 9 10
State variables of the LPV system
5
Estimated state variables by the Observer 5
Outputs of the LPV system
5
Outputs the LPV+Observer system
5
Figure 3.18.: States and outputs of the controlled LPV and controlled and observed LPV system
Figure3.20. shows the PS of the simple LPV system and the PS, which is realized by the observer. The order of the error between the scheduling variables are very low: 10−2, which means the completed observer approximates the scheduling variables, however, with high accuracy.
3.6.2. Summary
In the Section above, I have introduced an LPV-based completed observer design ap-proach. This method provides a mixture of classical state observer design and a kind of supplementary observer which is based on the parameter vectors (and belonging parameter dependent LTI systems) of the LPV parameter space.
Time [min]
0 1 2 3 4 5 6 7 8 9 10
State errors
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
x1,LP V(t)−ˆx1(t) x2,LP V(t)−ˆx2(t)
Time [min]
0 1 2 3 4 5 6 7 8 9 10
Output errors
×10-4
-6 -4 -2 0 2 4 6
y1,LPV(t)-y 1,LPV+OBS(t) y2,LPV(t)-y
2,LPV+OBS(t)
Figure 3.19.: Result of the simulations
I presented the properties, key points and usability of the developed completed observer scheme.
The developed LPV based completed observer structure is able to estimate the actual values of the states in case the states are directly not measurable. I presented the usability of the developed tools in case of nonlinear physiological system.
I found that the completed observer can accurately estimate the states of the given specific LPV system.
Finally, a crucial point should be noted at this point: the key issue in this Subsection is the invertibility of C. I demonstrated the usability of the completed observer method in a favorable case. However, the method is able to deal with such situations, whenC is invertible, but not a diagonal matrix. That means, the states do not occur by their own on the output, or the order of the output equations are different. In that situations, when there is a coupling, eg. y1(t) =x1(t) +x3(t), y2(t) =x3(t) andy3(t) =x2(t) the
0.6 0.7
Figure 3.20.: Parameter space and parameter errors. Upper row: PS of LPV system;
middle row: PS of the observer; lower row: parameter error
C=
. The completed observer structure can deal with such situations – however, the complex investigation of these circumstances will be a part of my future work.
Thesis Group 2
Thesis group 2: Completed LPV controller and observer scheme for LPV systems.
Thesis 2
I have introduced mathematical tools for LPV related control tasks which successfully exploit the possibilities lied in the specific prop-erties of the parameter space of LPV systems. By using these tools different quality markers can be defined and specific complementary LPV controller and observer structures can be designed.
Thesis 2.1
I have introduced a norm based ”difference” interpretation regard-ing the LPV systems, based on the properties of the LPV parameter space. I have defined how to use these interpretations as error and quality criteria during modeling and control and demonstrated my theoretical findings on a concrete example in diabetes modeling.
Thesis 2.2
I have developed an LPV based complementary controller struc-ture in order to control nonlinear systems. The developed method requires the knowledge of classical state feedback theorems and less complex than the LMI-based methods, moreover it requires less computational capacity than the LMI-based techniques. I have demonstrated the usability of the developed tool in case of different nonlinear systems, with unfavorable circumstances demonstrating that the developed method provides stability and appropriate con-trol action.
Thesis 2.3
I have developed an LPV based complementary observer structure which can estimate the actual values of the states in case of directly not measurable ones. I demonstrated the usability of the developed tools in case of a nonlinear system. I have proven that the com-plementary observer can accurately estimate the states of the given specific LPV systems.
Relevant own publications pertaining to this thesis group: [91,103,104,105].