State Observer (Estimator) Lecture 16: State Observer (Estimator)

Objectives Define the observability of dynamic systems. Design state estimators (observers). Select the eigenvalues of the observer.

Observability Definition: A system is observable if the knowledge of the input u(0), u(1), …, u(n-1) and the output y(0), y(1),…,y(n-1) is sufficient to determine the initial state x(0). Condition for observability: To derive the condition under which the system is observable, let us consider, for simplicity, the unforced system (no input):

Note that x(0) has n unknowns Note that x(0) has n unknowns. This requires n equations to have a unique solution. For single output systems, the matrix Δo, called the observability matrix, is of size n x n. For the system to be observable, Δo must be invertible or, more generally, be of full rank n.

Example 1 Given the following discrete-time system Check if the system is observable. This means that x(0) can not be uniquely estimated (calculated) from input-output data sequence. That is different initial states will result in the same output sequence.

State estimation In designing state feedback controller, we assumed that all states are measured. In most applications, measuring the entire state vector is impossible or prohibitively expensive. To implement state feedback control, an estimate of the state vector can be used. The state can be estimated from the input and output measurements by using a state estimator or observer. A state estimator is a mathematical construct (implemented as a part of the controller program) that helps estimate the states of a system.

Observer: simple idea Consider the following system: To estimate all the states, one could in theory use a system with the same state equation: Here, we assume perfect knowledge of the system dynamics (i.e. the matrices Φ, Γ).

Simple observer: drawback Let us examine the error dynamics of the simple observer by subtracting the observer dynamics from the system dynamics, i.e. The state error dynamics are determined by the state matrix Φ. This matrix cannot be chosen arbitrarily. For an unstable system, the observer will be unstable and cannot track the state of the system.

Observer: a better idea A practical alternative is to provide the estimator with the measured output in addition to the input as shown. This corrects the state estimate.

Observer: a better idea The observer will calculate the difference between the measured and the estimated output of the system and use this difference as additional input in its state equation as: Subtracting the observer state equation from the system dynamics yields the estimation error dynamics

Observer: a better idea The error dynamics are now governed by the eigenvalues of the observer matrix Φo. Thus, the gain matrix L can be designed so that Φo has eigenvalues with fast modes to guarantee fast convergence of the estimated state to the true state (i.e. to make the error in estimated state goes to zero quickly).

Theorem: State Estimation If the pair (Φ, C) is observable, then there exists a matrix L that arbitrarily assigns the observer poles (eigenvalues of the matrix Φo = Φ - LC) to any set {λi, i = 1, . . . , n}.

Selecting estimator pole locations Typically, estimator pole locations are selected so that the estimator responds two to six times faster than the controlled plant.

Example 2 Given the following discrete-time system Check if the system is observable and controllable. If so, design a state feedback controller as well as a state observer such that: The eigenvalues of the state feedback system is ½ and -½ The eigenvalues of the observer is ¼ and -¼.

Solution Check if the system is observable and controllable: The controllability matrix The observability marix

Solution: computing the state feedback gain matrix K

Solution: observer design (computing the matrix L)

Observer design using MATLB Given the observer state matrix We transpose the matrix to obtain By this way we can use the same MATLAB command “place” as follows Lt = place(phi’,C’,p); L = Lt’; % note that the output need to be transposed