1. A few illustrations on the Basic Concepts of Nonlinear Control
Chapter 2
A few illustrations on the Basic Concepts of Nonlinear Control

2. Contents Input-State Feedback linearization
Input-Output Feedback Linearization
Basic Concepts of Zero Dynamics
Partial Feedback Linearization
Integrator Backstepping
Conclusions

3. Input-State Feedback Linearization
Short comings of Jacobean linearization
Linearized model is valid only for small operating region near equilibrium.
Mainly useful, for piecewise linear system
Not suitable for dynamic environment
The above mentioned drawbacks can be avoided using a feedback linearzing controller.
Consider the state model of the second order nonlinear systems:
b is a nonzero constant system parameter.
Equilibrium point= (0,0)

4. Input-State Feedback Linearization
Careful observation reveals that the linearize model is only valid for
Here we assume that all the states are available for ready measurement.
However, one problem of the given state model is that the said model contains unmatched nonlinearity. More precisely, the nonlinear term that is present in the first equation is not spanned by the control input u.
Therefore, in order to apply the input-state feedback linearization the system must be transformed into a new state coordinate, where all the nonlinear terms are matched by the control input.
Let us define a new state transformation

5. Input-State Feedback Linearization
Hence,
Now, in z1 and z2 coordinate all the nonlinear terms are matched by control input u.
Therefore, it is easy to design a u that can cancel out all the nonlinear entries from the equation of
That yields
Poles of the system are located at -2, 0.

6. Input-State Feedback Linearization
Now any one can design a state feedback to place the pole at any desired position.
Hence, if we want that the closed system will behave like a linear system with damping ratio ζ=0.7, and natural frequency ωn=5rad/s, then the pole must be placed at ( j and j).
Therefore, a control input should be designed as follows:
Accordingly the input equation in X coordinate:

7. Input-Output Feedback Linearization
Short comings of Input-State Feedback Linearization
All the states may not be available for measurement.
More number of measurement induces more measuring error.
If only output from the system is available, then it is impossible to implement input state feedback linearization.
Hence, in such a case one should try to design an input-output feedback linearization.
Let us consider the following system

8. Input-Output Feedback Linearization
Since only output from the system is only measureable signal. One must try to establish an explicit relation between input and output signals.
A simple way to achieve an explicit relation between input and output is to differentiate the output until it exhibits any direct relationship with input.
Two successive differentiations yield an explicit relation between input and out as shown below:
Where
Now one can use the following control input to cancel out nonlinear terms

9. Input-Output Feedback Linearization
Application of the control input u yields a simple double integrator relationship between equivalent input v with the output y:
Now if the intended control objective is to track the output yd , then one can design a control law , and tracking error is defined as
Now if v is defined to be :
Such v yields
for all positive values of k2 and k1 ensure asymptotic stable error dynamics that yields perfect tracking.
Therefore, perfect tracking can be achieved in almost all cases except the singular point that is located at

10. Input-Output Feedback Linearization
Indeed, the input-output model (equation (xi)) imitates the dynamics of a second order system. However, order reduction of a dynamics system is not possible. Here in lies the importance of internal dynamics.
For an nth order system at the time of Input-Output Linearization, if r differentiation is required to establish an explicit relationship between input u and output y, then the system can be termed as a system with relative degree r (Isidori 1991) .
Therefore, it is easy to conclude that the system of equation (viii) has a relative degree equal to two.
Frequently a part of the system dynamics is rendered unobservable at the time of Input-Output Linearization. In generally, this part of the dynamics of the system is termed as the Internal Dynamics of the original system.
Stability of internal dynamics of the system can be assessed using the concept of zero dynamics.

11. Basic Concept of Zero Dynamics
Let us consider a simple 3rd order system transfer function
State Model Representation
Three consecutive differentiation of y yields explicit relation between input and output
Relative Degree of the system is THREE.

12. Basic Concept of Zero Dynamics
Relative degree indicates the excess number of poles over the number of zeros.
Order of the internal dynamics 3-3=0.
Now append one zero to that same transfer function
State Model

13. Basic Concept of Zero Dynamics
Two consecutive differentiation of y yields explicit relation between input and output
Relative degree of the system is 2.
Order of the internal dynamics is 1.
Now use of the following feedback law
yields

14. Basic Concept of Zero Dynamics
Therefore, if we consider the output y as the state of the system and signal v as the input to the system, then
Basically this is a second order state model, so we can conclude that application input , converts the original system into a 2nd order state model!!!!!
Actually, it converts the system into a cascade combination of reduced order system and internal dynamics.

15. Basic Concept of Zero Dynamics
Internal dynamics can be represented as
Stability of internal dynamics can be assessed by the location of the zero at s plane!!!!
Consider a nonlinear system
First order differentiation of y yields

16. Basic Concept of Zero Dynamics
Now application of input yields
Internal dynamics
Zero Dynamics can be found using the concept of output zeroing input.
Output identically equal to zero implies
Now that implies
Again
Therefore, zero dynamics equation is

17. Partial Feedback Linearization
However, in case of Underactuated Mechanical Systems conventional linearization fails to linearize the system.
Actually, it is not possible to obtain a complete linearized model for any underactuated system.
Mark. W. Spong has invented a new type of feedback linearization and coined it as Partial Feedback Linearization to obtain a somewhat linearize model of the underactuated system.
Consider the lagrangian model for a generic UMS system.

18. Partial Feedback Linearization
Now, in case of partial feedback linearization. Equation of passive joints yields:
Now, replacing the expression of in the equation of active joints
Therefore, in order to linearize the active joints, the following control law may be applied:
The control law yields:

19. Partial Feedback Linearization
Now,
The above equation yields the following equation for passive joints :
Therefore,
where

20. Partial Feedback Linearization
The above-mentioned partial feedback linearization is defined as collocated feedback linearization. That yields a linear model for active joints.
Now, application of a new control input
Results in:
In the above system, passive joints are linearized. This type of linearization is known as non-collocated linearization.

21. Integrator Backstepping
Stabilization Problem of Dynamical System
Design objective is to construct a control input u which ensures the regulation of the state variables x(t) and z(t), for all x(0) and z(0).
Equilibrium point: x=0, z=0
Design objective can be achieved by making the above mentioned equilibrium a GAS.

22. Integrator Backstepping
Block Diagram of the system:

23. Integrator Backstepping
First step of the design is to construct a control input for the scalar subsystem
z can be considered as a control input to the scalar subsystem
Construction of CLF for the scalar subsystem
Control Law:
But z is only a state variable, it is not the control input.

24. Integrator Backstepping
Only one can conclude the desired value of z as
Definition of Error variable e:
z is termed as the Virtual Control
Desired Value of z, αs(x) is termed as stabilizing function.
System Dynamics in ( x, e) Coordinate:

25. Integrator Backstepping
Modified Block Diagram
Feedback Control Law αs
Backstepping Signal -αs

26. Integrator Backstepping
So the signal αs(x) serve the purpose of feedback control law inside the block and “backstep” -αs(x) through an integrator.
Feedback loop with + αs(x)
Backstepping of Signal -αs(x)
Through integrator

27. Integrator Backstepping
Construction of CLF for the overall 2nd order system:
Derivative of Va
A simple choice of Control Input u is:
With this control input derivative of CLF becomes:

28. Integrator Backstepping
Consider the scalar nonlinear system
Control Law( using Feedback Linearization):
Resultant System:
Edurado D. Sontag Proposed a formula to avoid the Cancellation of these useful nonlinearities.
Not at all!!!!
This is an Useful Nonlinearity, it has an Stabilizing effect on the system.
is it essential to cancel out the term

29. Integrator Backstepping
Sontag's Formula:
Control Law (Sontag’s Formula):
Control Law (using Backstepping):
But this formula leads a complicated control input for intermediate values of x
So this control law avoids the cancellation of useful nonlinearities! For higher values of x
For large values of x, the control law becomes
u≈sinx

30. Conclusions
Input-State Feedback Linearization may be used to obtain a linearize model for a nonlinear system.
However, input-state feedback linearization requires availability of all the states of the system, which makes it inapt for practical applications.
Input-Output Linearization may be considered as a suitable alternative for input-state feedback linearization.
However, input-output feedback linearization sometimes convert the system in a cascade combination of two dynamics systems, where one system represents the dynamics of reduced order system that is visible from input- output relation. Other system act as a hidden dynamical system that is commonly known as internal dynamics.
Stability of internal dynamics is being analyzed using the analogy of zeros of linear systems, and thereby this method is defined as zero dynamics analysis.
However, in case of UMS both type of feedback linearization fail to generate a satisfactory result. Therefore, for UMS, Mark. W. Spong has invented a new linearization technique that is known as Partial Feedback Linearization [PFL].
Two types of partial feedback linearization are Collocated partial feedback Linearization, and Noncollocated partial Feedback linearization. Collocated PFL linearize the UMS with respect to active joints, whereas the noncollocated PFL linearize the passive joints.
However, main drawback of feedback linearization is that it cancel out useful nonlinearities from system dynamics.
Therefore, one can use integrator backstepping that does not cancel out useful nonlinearities.
Only one prerequisite of backstepping design is that it requires system model in strict feedback form.