1. EE611 Deterministic Systems Controllability and Observability of Continuous-time Systems Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Controllability Observability Example For the circuit below with current source u as input, voltage y as output, and state defined as the voltage on the capacitors as indicated. Derive the state-space model and discuss the ability to control (affect) the state from the input and observe the state from the output. u(t)u(t) + x 1 -+ x 2 - 11 11 1F +y(t)-+y(t)- 22

3. Controllability The state equation: is controllable if for any pair of states x(t 1 ) and x(t 0 ),  an input u(t) that drives state x(t 0 ) to x(t 1 ) in a finite time. If the system is controllable, then an input to transfer state x(t 0 ) to x(t 1 ) over the time interval [t 0, t 1 ] is given by: where

4. Conditions for Controllability For an n state and p input system: This system is controllable for t > 0 if any one of the equivalent conditions are met: 1. The n x n matrix W c is nonsingular for all t > 0 2. The n x np controllability matrix C has full row rank (n):

5. Conditions for Controllability 3. The n x (n+p) matrix [(A- I) B] has full row rank for every eigenvalue of A. 4. All eigenvalues of A have negative real parts, and the unique solution W c is positive definite. where W c is the controllability Gramian:

6. Controllability Examples Given systems matrix A, determine which input vectors result in controllability Given systems matrix A, determine which input matrices result in controllability

7. Controllability Examples Given system matrix A and input vector b determine an input to bring initial state x(0) = 0 to x(4) = [-.2.8.1]' Hint: Do not do by hand. Plot input from 0 to 4 s when finished. Also consider changing the state to the desired value over a shorter time interval and describe how the input changes. For example try x(0) = 0 to x(2) = [-.2.8.1]'

8. Controllability Index Consider an n state x p input system. The controllability matrix is given by: where B can be expressed as a concatenation of column vectors: Let  m be the number of linearly independent vectors in C associated with with column m. Then C has rank n if Then { } are called the controllability indices and the maximum value of the set is call the controllability index.

9. Controllability and Equivalence Transformations The controllability property is invariant under under any equivalence transformation and reordering of columns of B. This can be shown by substituting similarity/equivalence transformation matrices P and Q to establish the equality below: And since P is nonsingular, its product with controllability matrix has the same rank as the controllability matrix.

10. Observability The state equation: is observable if for any unknown initial state x(t 0 ), there exists a finite time t 1 – t 0  knowledge of input u(t) and output input y(t) over [t 0, t 1 ] is all that is required to uniquely determine x(t 0 ). If the system is observable, an estimator/observer to compute state x(t 0 ) is given by:

11. Conditions for Observability For an n state and q output system: 1. This system is observable for t > 0 iff the nxn matrix W o is nonsingular for all t > 0. 2. The nq x n observability matrix O has full column rank (n):

12. Conditions for Observability 3. The (n+q) x n matrix [(A- I)' C']' has full column rank for every eigenvalue of A. 4. All eigenvalues of A have negative real parts, and the unique solution W o is positive definite where W o is the observability Gramian:

13. Observability Example Given system matrix A and input vector b o, output vector c o, with unit step input and x(0) = [-1, 5, 0]', estimate this initial state from the input and observing the output over interval t=[0 4] seconds. Hint: Do not do by hand. Verify that observer did indeed estimate the proper state value.

14. Observability Index Consider an n state x q output system. The observability matrix is given by: where C can be expressed as a concatenation of row vectors: Let  m be the number of linearly independent vectors in O associated with with row m. Then O has rank n if Then { } are called the observability indices and the maximum value of the set is call the observability index.

15. Observability and Equivalence Transformations The observability property is invariant under under any equivalence transformation and any reordering of rows of C. Note: Rows k and m of an nxn matrix A can be interchanged by pre-multiplying A by the identity matrix with the k th and m th rows interchanged. For example to interchange rows 2 and 4 of a 5x5 matrix, multiply by: To interchange columns 2 and 4 post-multiply by the same matrix.

16. Lecture Note Homework U8.1 Given system matrix A and input vector b plot an input required to bring initial state x(0) = 0 to x(10) = [-1, 1, -1]' and compute the energy of the input. Repeat the previous problem for a final state of x(10) = [-0.1, 0.1, -0.1]' Do you see a relationship between the distance between the states and the required energy? Hand in a printout of the commented code used in addition to the plots and results..

17. Lecture Note Homework U8.2 Given system matrix A, input vector b, output vector c, input a unit step, and output for t  0. Determine the original state at t=0. Hand in a printout of the commented code used in addition to the result.