In this section, we demonstrate the efficacy of the feedback linearizable intervention approach described in this paper by applying it to a well-studied biological pathway model representing the glycolytic-glycogenolytic pathway shown in [

17,

29]. Glycolysis is the process of breaking up a six-carbon glucose molecule into two molecules of a three-carbon compound, and glycogenolysis is the process by which the stored glycogen in the body is broken up to meet the needs for glucose. In glycogenolysis, the phosphorylase enzyme acts on the polysaccharide glycogen to reduce its length by one glucose unit. The glucose unit is released as a glucose-1 phosphate. The glycolytic-glycogenolytic pathway can be mathematically represented by the following S-system model:

In this case, *N* = 3, *m* = 7 and the parameter are defined as *α*_{1} = 0.077884314, *θ*_{14} = 0.66, *θ*_{16} = 1, *β*_{1} = 1.06270825, *μ*_{11} = 1.53, *μ*_{12} = −0.59, *μ*_{17} = 1, *α*_{2} = 0.585012402, *θ*_{21} = 0.95, *θ*_{22} = −0.41, *θ*_{25} = 0.32, *θ*_{27} = 0.62, *θ*_{210} = 0.38, *β*_{2} = *α*_{3} = 0.0007934561, *μ*_{22} = *θ*_{32} = 3.97, *μ*_{23} = *θ*_{33} = −3.06, *μ*_{28} = *θ*_{38} = 1, *β*_{3} = 1.05880847, *μ*_{33} = 0.3, and *μ*_{39} = 1. Here, the model variables are defined as follows: *x*_{1} is glucose-1-P, *x*_{2} is glucose-6-P, *x*_{3} is fructose-6-P, *x*_{4} is inorganic phosphate ion, *x*_{5} is glucose, *x*_{6} is phosphorylase *a*, *x*_{7} is phosphoglucomutase, *x*_{8} is phosphoglucose isomerase, *x*_{9} is phosphofructokinase, and *x*_{10} is glucokinase.

For this model, the metabolites

*x*_{4} through

*x*_{10} are defined as independent variables, which are the variables that are not affected by other variables, and the metabolites

*x*_{1} through

*x*_{3} are defined as the dependent variables, which are the primary variables of interest that we wish to control. Here, we choose the independent variables

*x*_{4},

*x*_{5}, and

*x*_{8} as manipulated or control variables, as shown in , as they can affect the production of the dependent variables

*x*_{1},

*x*_{2}, and

*x*_{3}. Also, we choose to keep the independent variables

*x*_{6},

*x*_{7},

*x*_{9}, and

*x*_{10} fixed ignoring their effect on the controlled variables, and assuming that the controller only uses the independent variables

*x*_{4},

*x*_{5}, and

*x*_{8} to control the dependent variables

*x*_{1},

*x*_{2}, and

*x*_{3}. The independent variables have the following values

*x*_{4} = 10,

*x*_{5} = 5,

*x*_{6} = 3,

*x*_{7} = 40,

*x*_{8} = 136,

*x*_{9} = 2.86, and

*x*_{10} = 4. Here, we try to control

*x*_{1},

*x*_{2}, and

*x*_{3} by manipulating

*x*_{4},

*x*_{5}, and

*x*_{8}, so we have

and all other

*x*_{i}′

*s* for

*i* = 6,7, 9, and 10 are kept fixed. The initial values of the outputs

*y*_{1},

*y*_{2}, and

*y*_{3} are selected as 0.067, 0.465, and 0.150, respectively, and the desired reference outputs are selected as

*y*_{d1} = 0.2,

*y*_{d2} = 0.5, and

*y*_{d3} = 0.4.

Hence, the overall system can be expressed in the form of (

6), where

where

*a*_{1} =

*α*_{1}*x*_{6}^{θ16},

*a*_{2} =

*α*_{2}*x*_{7}^{θ27}*x*_{10}^{θ210},

*a*_{3} =

*α*_{3},

*b*_{1} =

*β*_{1}*x*_{7}^{μ17},

*b*_{2} =

*β*_{2}, and

*b*_{3} =

*β*_{3}*x*_{9}^{μ39}.

Based on the S-system model describing the glycolytic-glycogenolytic pathway, it can be verified that the outputs need to be differentiated twice with respect to time so that the input variables (

*u*_{1},

*u*_{2}, or

*u*_{3}) appear in the expressions of differentiated outputs, as follows:

where

Hence, in this case the system has vector relative degree

*γ* = [

*γ*_{1},

*γ*_{2},

*γ*_{3}]

^{T} = [2,2,2]

^{T}, and hence we have

*γ*_{1} +

*γ*_{2} +

*γ*_{3} = 6.

The matrix form of the system of differential equations presented in (

25) can be written in the form of (

14), where

The matrix

*D*(

*x*) is invertible if the following condition is satisfied:

Based on (

25), it can be seen that the control variables

*u*_{1} and

*u*_{2} appear only in the expressions of

*y*_{1}^{(2)} and

*y*_{2}^{(2)}, respectively. However,

*u*_{3} appears in the expressions of

*y*_{2}^{(2)} and

*y*_{3}^{(2)}. Hence,

*u*_{1} and

*u*_{3} need to be used to control

*y*_{1} and

*y*_{3}, respectively, and both

*u*_{2} and

*u*_{3} are needed to control

*y*_{2}.

Hence, the control laws based on (

16) can be expressed as

Substituting the expressions of the control variables (

29) in (

25), we obtain the following decoupled linear system:

The new control variables

*v*_{j}, for

*j* = 1,2, 3, need to be designed so that the target variables

*y*_{j} track some desired reference trajectories,

*y*_{dj}.

Using (

20), the new control variables

*v*_{j}, for

*j* = 1,2, 3, are found to be

The new control components,

*v*_{1},

*v*_{2}, and

*v*_{3}, are defined in (

31), where the parameters are selected as

*k*_{1} = 1,

*k*_{11} = 5,

*k*_{2} = 10

^{−3},

*k*_{21} = 20,

*k*_{3} = 3, and

*k*_{31} = 5.

Figures and show the output response and the control input signals when the feedback linearizable controller is applied. It is clear from that the system outputs converge to their desired values. Another simulation study is implemented for a different reference trajectory, where the value of the reference signal increases linearly before saturating at the desired final value. The closed-loop output response in this case is shown in and the control signals are shown in . It is clear from that the feedback linearizable controller is driving the target variables to track the desired reference trajectories.

To study the robustness properties of the feedback linearizable controller, similar simulation studies have been conducted when the parameters *μ*_{22} and *β*_{2} are varied within 10% of their nominal values. It has been found that the closed-loop system is stable only for parameter variations within 1% and with unacceptable performance. This agrees with our earlier assumption that full system knowledge is needed for proper operation of the feedback linearizable controller.