.. currentmodule:: control

This section describes the functions the are available to analyze state space systems and design state feedback controllers. The functionality described here is mainly specific to state space system representations; additional functions for analysis of linear input/output systems, including transfer functions and frequency response data systems, are defined in the next section and can also be applied to LTI systems in state space form.

The following basic attributes and methods are available for :class:`StateSpace` objects:

.. autosummary::

   ~StateSpace.A
   ~StateSpace.B
   ~StateSpace.C
   ~StateSpace.D
   ~StateSpace.dt
   ~StateSpace.shape
   ~StateSpace.nstates
   ~StateSpace.poles
   ~StateSpace.zeros
   ~StateSpace.dcgain
   ~StateSpace.sample
   ~StateSpace.returnScipySignalLTI
   ~StateSpace.__call__

A complete list of attributes, methods, and properties is available in the :class:`StateSpace` class documentation.

State space systems can be transformed into different internal representations representing a variety of standard canonical forms that have the same input/output properties. The :func:`similarity_transform` function allows a change of internal state variable via similarity transformation and the :func:`canonical_form` function converts systems into different canonical forms. Additional information is available on the documentation pages for the individual functions:

.. autosummary::

   canonical_form
   observable_form
   modal_form
   reachable_form
   similarity_transform

The following functions are available to analyze the time domain properties of a linear system:

.. autosummary::

   damp
   forced_response
   impulse_response
   initial_response
   ssdata
   step_info
   step_response

The time response functions (:func:`impulse_response`, :func:`initial_response`, :func:`forced_response`, and :func:`step_response`) are described in more detail in the :ref:`response-chapter` chapter.

State feedback controllers for a linear system are controllers of the form

u = -K x

where K \in {\mathbb R}^{m \times n} is a matrix of feedback gains. Assuming the systems is controllable, the resulting closed loop system will have dynamics matrix A - B K with stable eigenvalues.

Feedback controllers can be designed using one of several methods:

.. autosummary::

   lqr
   place
   place_acker
   place_varga

The :func:`place`, :func:`place_acker`, and :func:`place_varga` functions place the eigenvalues of the closed loop system to a desired set of values. Each takes the A and B matrices of the state space system and the desired location of the eigenvalues and returns a gain matrix K:

K = ct.place(sys.A, sys.B, E)

where E is a 1D array of desired eigenvalues.

The :func:`lqr` function computes the optimal state feedback controller that minimizes the quadratic cost

J = \int_0^\infty (x' Q x + u' R u + 2 x' N u) dt

by solving the appropriate Riccati equation. It returns the gain matrix K, the solution to the Riccati equation S, and the location of the closed loop eigenvalues E. It can be called in one of several forms:

• K, S, E = ct.lqr(sys, Q, R)
• K, S, E = ct.lqr(sys, Q, R, N)
• K, S, E = ct.lqr(A, B, Q, R)
• K, S, E = ct.lqr(A, B, Q, R, N)

If sys is a discrete-time system, the first two forms will compute the discrete-time optimal controller. For the second two forms, the :func:`dlqr` function can be used to compute the discrete-time optimal controller. Additional arguments and details are given on the :func:`lqr` and :func:`dlqr` documentation pages.

State estimators (or observers) are dynamical systems that estimate the state of a system given a model of the dynamics and the input and output signals as a function of time. Linear state estimators have the form

\frac{d\hat x}{dt} = A \hat x + B u + L(y - C\hat x - D u),

where \hat x is an estimate of the state and L \in {\mathbb R}^{n \times p} represents the estimator gain. The gain L is chosen such that the eigenvalues of the matrix A - L C are stable, resulting in an estimate that converges to the value of the system state.

The gain matrix L can be chosen using eigenvalue placement by calling the :func:`place` function:

L = ct.place(sys.A.T, sys.C.T, E).T

where E is the desired location of the eigenvalues and .T computes the transpose of a matrix.

More sophisticated estimators can be constructed by modeling noise and disturbances as stochastic signals generated by a random process. Estimators constructed using these models are described in more detail in the :ref:`kalman-filter` section of the :ref:`stochastic-systems` chapter.