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

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

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)

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

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.

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 (-3.5+3.75j and -3.5-3.75j). Therefore, a control input should be designed as follows: Accordingly the input equation in X coordinate:

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

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

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 .

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.

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.

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

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

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.

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

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

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.

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:

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

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.

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.

Integrator Backstepping Block Diagram of the system:

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.

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:

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

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

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:

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

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

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.