In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. "State space" refers to the Euclidean space in which the variables on the axes are the state variables. The state of the system can be represented as a vector within that space.
Contents
- State variables
- Linear systems
- Example continuous time LTI case
- Controllability
- Observability
- Transfer function
- Canonical realizations
- Proper transfer functions
- Feedback
- Example
- Feedback with setpoint reference input
- Moving object example
- Nonlinear systems
- Pendulum example
- References
To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. Additionally, if the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general system theory, which makes it possible to use Kronecker vector-matrix structures. The capacity of these structures can be efficiently applied to research systems with modulation or without it. The state-space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With
State variables
The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system,
Linear systems
The most general state-space representation of a linear system with
where:
In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in the common LTI case, matrices will be time invariant. The time variable
Example: continuous-time LTI case
Stability and natural response characteristics of a continuous-time LTI system (i.e., linear with matrices that are constant with respect to time) can be studied from the eigenvalues of the matrix A. The stability of a time-invariant state-space model can be determined by looking at the system's transfer function in factored form. It will then look something like this:
The denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of
The roots of this polynomial (the eigenvalues) are the system transfer function's poles (i.e., the singularities where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is asymptotically stable or marginally stable. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability.
The zeros found in the numerator of
The system may still be input–output stable (see BIBO stable) even though it is not internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if those singularities in the transfer function are removable).
Controllability
State controllability condition implies that it is possible – by admissible inputs – to steer the states from any initial value to any final value within some finite time window. A continuous time-invariant linear state-space model is controllable if and only if
where rank is the number of linearly independent rows in a matrix, and where n is the number of state variables.
Observability
Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals (i.e., as controllability provides that an input is available that brings any initial state to any desired final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the system).
A continuous time-invariant linear state-space model is observable if and only if
Transfer function
The "transfer function" of a continuous time-invariant linear state-space model can be derived in the following way:
First, taking the Laplace transform of
yields
Next, we simplify for
and thus
Substituting for
The transfer function
comparison with the equation for
Clearly
Canonical realizations
Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system):
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
The coefficients can now be inserted directly into the state-space model by the following approach:
This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
The transfer function coefficients can also be used to construct another type of canonical form
This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
Proper transfer functions
Transfer functions which are only proper (and not strictly proper) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant.
The strictly proper transfer function can then be transformed into a canonical state-space realization using techniques shown above. The state-space realization of the constant is trivially
Here is an example to clear things up a bit:
which yields the following controllable realization
Notice how the output also depends directly on the input. This is due to the
Feedback
A common method for feedback is to multiply the output by a matrix K and setting this as the input to the system:
becomes
solving the output equation for
The advantage of this is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of
Example
For a strictly proper system D equals zero. Another fairly common situation is when all states are outputs, i.e. y = x, which yields C = I, the Identity matrix. This would then result in the simpler equations
This reduces the necessary eigendecomposition to just
Feedback with setpoint (reference) input
In addition to feedback, an input,
becomes
solving the output equation for
One fairly common simplification to this system is removing D, which reduces the equations to
Moving object example
A classical linear system is that of one-dimensional movement of an object. Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring
where
The state equation would then become
where
The controllability test is then
which has full rank for all
The observability test is then
which also has full rank. Therefore, this system is both controllable and observable.
Nonlinear systems
The more general form of a state-space model can be written as two functions.
The first is the state equation and the latter is the output equation. If the function
Pendulum example
A classic nonlinear system is a simple unforced pendulum
where
The state equations are then
where
Instead, the state equation can be written in the general form
The equilibrium/stationary points of a system are when
for integers n.