Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Observers/Estimators …  bnbn b n-1 b2b2 b1b1 D anan a n-1 a2a2 a1a1  … … u z2z2 z n-1 z1z1  znzn y

Outline of Today’s Lecture Review Control System Objective Design Structure for State Feedback State Feedback 2 nd Order Response State Feedback using the Reachable Canonical Form Observability Observability Matrix Observable Canonical Form Use of Observers/Estimators

Control System Objective Given a system with the dynamics and the output Design a linear controller with a single input which is stable at an equilibrium point that we define as

Our Design Structure  Disturbance Controller Plant/Process Input r Output y x  -K krkr State Feedback Prefilter State Controller u

2 nd Order Response As the example showed, the characteristic equation for which the roots are the eigenvalues allow us to design the reachable system dynamics When we determined the natural frequency and the damping ration by the equation we actually changed the system modes by changing the eigenvalues of the system through state feedback  1 Re( ) Im( ) x x x x x x x x x x           n =1 nn 1 Re( ) Im( ) x x  x x x x  n =1  n =2  n =4

State Feedback Design with the Reachable Canonical Equation Since the reachable canonical form has the coefficients of the characteristic polynomial explicitly stated, it may be used for design purposes:

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 Since observability is a function of the dynamics, consider the following system without input: The output is Using the truncated series

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

Example Using the electric motor developed in Lecture 5, develop the Observability Canonical form using the values

Observers / Estimators Knowing that the system is observable, how do we observe the states? Observer/Estimator Input u(t)Output y(t) Noise State

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

Observers / Estimators The form of our observer/estimator is If (A-LC) has negative real parts, it is both stable and the error,, will go to zero. How fast? Depends on the eigenvaluesof (A-LC)

Observers / Estimators To compute L in we need to compute the observable canonical form with

Example A hot air balloon has the following equilibrium equations Construct a state observer assuming that the eigenvalute to achieve are =10:  h w u

Example

Control with Observers Previously we designed a state feedback controller where we generated the input to the system to be controlled as When we did that we assumed that wse had direct access to the states. But what if we do not? A possible solution is to use the observer/estimator states and generate

Control with Observers -K + + krkr B B C C A A L _ u y r

Control with Observers

Designing Controllers with Observers

Example A hot air balloon has the following equilibrium equations Construct a state feedback controller with an observer to achieve and maintain a given height  h w u

Example

Summary Observability 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. Observability Matrix Observable Canonical Form Use of Observers/Estimators Next: Kalman Filters