Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Transfer Functions

Outline of Today’s Lecture A new way of representing systems Derivation of the gain transfer function Coordinate transformation effects hint: there are none! Development of the Transfer Function from an ODE Gain, Poles and Zeros

Observability Can we determine what are the states that produced a certain output? Perhaps Consider the linear system We say the system is observable if for any time T>0 it is possible to determine the state vector, x(T), through the measurements of the output, y(t), as the result of input, u(t), over the period between t=0 and t=T.

Observers / Estimators Observer/Estimator Input u(t)Output y(t) Noise State

Testing for Observability For x(0) to be uniquely determined, the material in the parens must exist requiring to have full rank, thus also being invertible, the common test W o is called the Observability Matrix

Testing for Observability For x(0) to be uniquely determined, the material in the parens must exist requiring to have full rank, thus also being invertible, the common test W o is called the Observability Matrix

Example: Inverted Pendulum Determine the observability pf the Segway system with v as the output

Observable Canonical Form A system is in Observable Canonical Form if it can be put into the form …  bnbn b n-1 b2b2 b1b1 D anan a n-1 a2a2 a1a1  … … u z2z2 z n-1 z1z1  znzn y Where a i are the coefficients of the characteristic equation

Observable Canonical Form

Dual Canonical Forms

Observers / Estimators B B C C A A L _ u y Observer/Estimator Input u(t) Output y(t) Noise State

Alternative Method of Analysis Up to this point in the course, we have been concerned about the structure of the system and discribed that structure with a state space formulation Now we are going to analyze the system by an alternative method that focuses on the inputs, the outputs and the linkages between system components. The starting point are the system differential equations or difference equations. However this method will characterize the process of a system block by its gain, G(s), and the ratio of the block output to its input. Formally, the transfer function is defined as the ratio of the Laplace transforms of the Input to the Output:

System Response From Lecture 11 We derived for } } Transient Steady State Transfer function is defined as

Derivation of Gain Consider an input of The first term is the transient and dies away if A is stable.

Example m k x u(t) b

Example m k x u(t) b

Coordination Transformations Thus the Transfer function is invariant under coordinate transformation x1x1 x2x2 z2z2 z1z1

Linear System Transfer Functions General form of linear time invariant (LTI) system is expressed: For an input of u(t)=e st such that the output is y(t)=y(0)e st Note that the transfer function for a simple ODE can be written as the ratio of the coefficients between the left and right sides multiplied by powers of s The order of the system is the highest exponent of s in the denominator.

Simple Transfer Functions Differential Equation Transfer Function Name s Differentiator Integrator 2 nd order Integrator 1 st order system Damped Oscillator PID Controller

A Different Method Design a controller that will control the angular position to a given angle,  0

A Different Method Design a controller that will control the angular position to a given angle,  0

A Different Method R=0.2 J= 10^-5 K=6*10^-5 Kb=5.5*10^-2 b=4*10^-2 gs=(K/(J*R))/(b*s^2/J+K*Kb*s/(J*R)) step(gs) Design a controller that will control the angular position to a given angle,  0 Which was the same for the state space Later, we will learn how to control it

Gain, Poles and Zeros The roots of the polynomial in the denominator, a(s), are called the “poles” of the system The poles are associated with the modes of the system and these are the eigenvalues of the dynamics matrix in a state space representation The roots of the polynomial in the numerator, b(s) are called the “zeros” of the system The zeros counteract the effect of a pole at a location The value of G(s) is the zero frequency or steady state gain of the system

Summary A new way of representing systems The transfer function Derivation of the gain transfer function Coordinate transformation effects hint: there are none! Development of the Transfer Function from an ODE Gain, Poles and Zeros Next: Block Diagrams