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Encyclopedia > State space (controls)

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. To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the last one can be done when the dynamical system is linear and time invariant). 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 p inputs and q outputs, we would otherwise have to write down q times p Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space. Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

Contents

State variables

Typical state space model
Typical state space model

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. State variables must be linearly independent; a state variable cannot be a linear combination of other state variables. The minimum number of state variables required to represent a given system, n, is usually equal to the order of the system's defining differential equation. If the system is represented in transfer function form, the minimum number of state variables is equal to the transfer function's denominator after it has been reduced to a proper fraction. It is important to understand that converting a state space realization to a transfer function form may lose some internal information about the system, and may provide a description of a system which is stable, when the state-space realization is unstable at certain points. In electronic systems, the number of state variables is the same as the number of energy storage elements in the circuit (capacitors and inductors). Typical State space (controls) model File links The following pages link to this file: State space (controls) User:Cburnett/Images Categories: User-created public domain images ... A state variable is any variable which represents the state of an object. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ...


Linear systems

The most general state space representation of a linear system with p inputs, q outputs and n state variables is written in the following form:

dot{mathbf{x}}(t) = A(t) mathbf{x}(t) + B(t) mathbf{u}(t)
mathbf{y}(t) = C(t) mathbf{x}(t) + D(t) mathbf{u}(t)

where

x(t) in mathbb{R}^n; y(t) in mathbb{R}^q; u(t) in mathbb{R}^p;
operatorname{dim}[A(cdot)] = n times n,
operatorname{dim}[B(cdot)] = n times p,
operatorname{dim}[C(cdot)] = q times n,
operatorname{dim}[D(cdot)] = q times p,
dot{mathbf{x}}(t) := {dmathbf{x}(t) over dt}.

x(cdot) is called the "state vector", y(cdot) is called the "output vector", u(cdot) is called the "input (or control) vector", A(cdot) is the "state matrix", B(cdot) is the "input matrix", C(cdot) is the "output matrix", and D(cdot) is the "feedthrough (or feedforward) matrix". For simplicity, D(cdot) is often chosen to be the zero matrix, i.e. the system is chosen not to have direct feedthrough. Notice that in this general formulation all matrixes are supposed to be time-variant, i.e. some or all their elements can depend on time. The time variable t can be a "continuous" one (i.e. t in mathbb{R}) or a discrete one (i.e. t in mathbb{Z}): in the latter case the time variable is usually indicated as k. Depending on the assumptions taken, the state-space model representation can assume the following forms:

System type State-space model
Continuous time-invariant dot{mathbf{x}}(t) = A mathbf{x}(t) + B mathbf{u}(t)
mathbf{y}(t) = C mathbf{x}(t) + D mathbf{u}(t)
Continuous time-variant dot{mathbf{x}}(t) = mathbf{A}(t) mathbf{x}(t) + mathbf{B}(t) mathbf{u}(t)
mathbf{y}(t) = mathbf{C}(t) mathbf{x}(t) + mathbf{D}(t) mathbf{u}(t)
Discrete time-invariant mathbf{x}(k+1) = A mathbf{x}(k) + B mathbf{u}(k)
mathbf{y}(k) = C mathbf{x}(k) + D mathbf{u}(k)
Discrete time-variant mathbf{x}(k+1) = mathbf{A}(k) mathbf{x}(k) + mathbf{B}(k) mathbf{u}(k)
mathbf{y}(k) = mathbf{C}(k) mathbf{x}(k) + mathbf{D}(k) mathbf{u}(k)
Laplace domain of
continuous time-invariant
s mathbf{X}(s) = A mathbf{X}(s) + B mathbf{U}(s)
mathbf{Y}(s) = C mathbf{X}(s) + D mathbf{U}(s)
Z-domain of
discrete time-invariant
z mathbf{X}(z) = A mathbf{X}(z) + B mathbf{U}(z)
mathbf{Y}(z) = C mathbf{X}(z) + D mathbf{U}(z)

Stability and natural response characteristics of a system can be studied from the eigenvalues of the matrix A. The stability of a time-invariant state-space model can easiest be determined by looking at the system's transfer function in factored form. It will then look something like this: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

 textbf{G}(s) = k frac{ (s - z_{1})(s - z_{2})(s - z_{3}) }{ (s - p_{1})(s - p_{2})(s - p_{3})(s - p_{4}) }

The denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

λ(s) = | sIA | .

The roots of this polynomial (the eigenvalues) yield the poles in the system's transfer function. 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 textbf{G}(s) can similarly be used to determine whether the system is minimum phase. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In the theory of dynamical systems, a linear time-invariant system is marginally stable if every eigenvalue in the systems transfer-function is non-positive, and all eigenvalues with zero real value are simple roots. ... In mathematics, the notion of Lyapunov stability occurs in the study of dynamical systems. ... In control theory and signal processing, a linear, time-invariant system is minimum-phase if the system and its inverse are causal and stable. ...


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. In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ...


Controllability and observability

A continuous time-invariant state-space model is controllable if and only if IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

operatorname{rank}begin{bmatrix}B& AB& A^{2}B& ...& A^{n-1}Bend{bmatrix} = n

A continuous time-invariant state-space model is observable if and only if

operatorname{rank}begin{bmatrix}C CA ... CA^{n-1}end{bmatrix} = n

(Rank is the number of linearly independent rows in a matrix.) In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...


See also: controllability and observability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. ... Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. ...


Transfer function

The "transfer function" of a continuous time-invariant state-space model can be derived in the following way: A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...


First, taking the laplace transform of In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

dot{mathbf{x}}(t) = A mathbf{x}(t) + B mathbf{u}(t)

yields

smathbf{X}(s) = A mathbf{X}(s) + B mathbf{U}(s)

Next, we simplify for mathbf{X}(s), giving

(smathbf{I} - A)mathbf{X}(s) = Bmathbf{U}(s)
mathbf{X}(s) = (smathbf{I} - A)^{-1}Bmathbf{U}(s)

this is substituted for mathbf{X}(s) in the output equation

mathbf{Y}(s) = Cmathbf{X}(s) + Dmathbf{U}(s), giving
mathbf{Y}(s) = C((smathbf{I} - A)^{-1}Bmathbf{U}(s)) + Dmathbf{U}(s)

Since the transfer function mathbf{G}(s) is defined as the ratio of the output to the input of a system, we take A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

mathbf{G}(s) = mathbf{Y}(s) / mathbf{U}(s)

and substite the previous expression for mathbf{Y}(s) with respect to mathbf{U}(s), giving

mathbf{G}(s) = C(smathbf{I} - A)^{-1}B + D

Clearly mathbf{G}(s) must have q by p dimensionality, and thus has a total of qp elements. So for every input there are q transfer functions with one for each output. This is why the state-space representation can easily be the preferred choice for multiple-input, multiple-output (MIMO) systems.


Canonical realizations

Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach: Strictly proper in control theory Strictly proper denotes a transfer function where the degree of the numerator is less than the degree of the denominator. ...


Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:

 textbf{G}(s) = frac{n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}.

The coefficients can now be inserted directly into the state-space model by the following approach:

dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& -d_{2}& -d_{3}& -d_{4} 1& 0& 0& 0 0& 1& 0& 0 0& 0& 1& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} 1 0 0 0 end{bmatrix}textbf{u}(t)
 textbf{y}(t) = begin{bmatrix} n_{1}& n_{2}& n_{3}& n_{4} end{bmatrix}textbf{x}(t).

This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable.


The transfer function coefficients can also be used to construct another type of canonical form

dot{textbf{x}}(t) = begin{bmatrix} -d_{1}& 1& 0& 0 -d_{2}& 0& 1& 0 -d_{3}& 0& 0& 1 -d_{4}& 0& 0& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} n_{1} n_{2} n_{3} n_{4} end{bmatrix}textbf{u}(t)
 textbf{y}(t) = begin{bmatrix} 1& 0& 0& 0 end{bmatrix}textbf{x}(t).

This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable.


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. Proper in control theory Proper denotes a transfer function where the degree of the numerator does not exceed the degree of the denominator. ... Strictly proper in control theory Strictly proper denotes a transfer function where the degree of the numerator is less than the degree of the denominator. ...

 textbf{G}(s) = textbf{G}_{SP}(s) + textbf{G}(infty)

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 textbf{y}(t) = textbf{G}(infty)textbf{u}(t). Together we then get a state space realization with matrices A,B and C determined by the strictly proper part, and matrix D determined by the constant.



Here is an example to clear things up a bit:

 textbf{G}(s) = frac{s^{2} + 3s + 3}{s^{2} + 2s + 1} = frac{s + 2}{s^{2} + 2s + 1} + 1

which yields the following controllable realization

dot{textbf{x}}(t) = begin{bmatrix} -2& -1 1& 0 end{bmatrix}textbf{x}(t) + begin{bmatrix} 1 0end{bmatrix}textbf{u}(t)
 textbf{y}(t) = begin{bmatrix} 1& 2end{bmatrix}textbf{x}(t) + begin{bmatrix} 1end{bmatrix}textbf{u}(t)

Notice how the output also depends directly on the input. This is due to the textbf{G}(infty) constant in the transfer function.


Feedback

Typical state space model with feedback
Typical state space model with feedback

A common method for feedback is to multiply the output by a matrix K and setting this as the input to the system: mathbf{u}(t) = K mathbf{y}(t). Since the values of K are unrestricted the values can easily be negated for negative feedback. The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results. Typical State space (controls) model with feedback File links The following pages link to this file: State space (controls) User:Cburnett/Images Categories: User-created public domain images ... This article does not cite any references or sources. ...

dot{mathbf{x}}(t) = A mathbf{x}(t) + B mathbf{u}(t)
mathbf{y}(t) = C mathbf{x}(t) + D mathbf{u}(t)

becomes

dot{mathbf{x}}(t) = A mathbf{x}(t) + B K mathbf{y}(t)
mathbf{y}(t) = C mathbf{x}(t) + D K mathbf{y}(t)

solving the output equation for mathbf{y}(t) and substituting in the state equation results in

dot{mathbf{x}}(t) = left(A + B K left(I - D Kright)^{-1} C right) mathbf{x}(t)
mathbf{y}(t) = left(I - D Kright)^{-1} C mathbf{x}(t)

The advantage of this is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of left(A + B K left(I - D Kright)^{-1} C right). This assumes that the open-loop system is controllable or that the unstable eigenvalues of A can be made stable through appropriate choice of K. In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. ...


One fairly common simplification to this system is removing D and setting C to identity, which reduces the equations to

dot{mathbf{x}}(t) = left(A + B K right) mathbf{x}(t)
mathbf{y}(t) = mathbf{x}(t)

This reduces the necessary eigendecomposition to just A + BK.


Feedback with setpoint (reference) input

State feedback with set point
State feedback with set point

In addition to feedback, an input, r(t), can be added such that mathbf{u}(t) = -K mathbf{y}(t) + mathbf{r}(t). Typical State space (controls) model with feedback and input File links The following pages link to this file: State space (controls) Categories: User-created public domain images ...

dot{mathbf{x}}(t) = A mathbf{x}(t) + B mathbf{u}(t)
mathbf{y}(t) = C mathbf{x}(t) + D mathbf{u}(t)

becomes

dot{mathbf{x}}(t) = A mathbf{x}(t) - B K mathbf{y}(t) + B mathbf{r}(t)
mathbf{y}(t) = C mathbf{x}(t) - D K mathbf{y}(t) + D mathbf{r}(t)

solving the output equation for mathbf{y}(t) and substituting in the state equation results in

dot{mathbf{x}}(t) = left(A - B K left(I + D Kright)^{-1} C right) mathbf{x}(t) + B left(I - K left(I + D Kright)^{-1}D right) mathbf{r}(t)
mathbf{y}(t) = left(I + D Kright)^{-1} C mathbf{x}(t) + left(I + D Kright)^{-1} D mathbf{r}(t)

One fairly common simplification to this system is removing D, which reduces the equations to

dot{mathbf{x}}(t) = left(A - B K C right) mathbf{x}(t) + B mathbf{r}(t)
mathbf{y}(t) = C mathbf{x}(t)

Moving object example

A classical linear system is that of one-dimensional movement of an object. The Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

m ddot{y}(t) = u(t) - k_1 dot{y}(t) - k_2 y(t)

where

  • y(t) is position; dot y(t) is velocity; ddot{y}(t) is acceleration
  • u(t) is an applied force
  • k1 is the viscous friction coefficient
  • k2 is the spring constant
  • m is the mass of the object

The state equation would then become

left[ begin{matrix} mathbf{dot{x_1}}(t)  mathbf{dot{x_2}}(t) end{matrix} right] = left[ begin{matrix} 0 & 1  -frac{k_2}{m} & -frac{k_1}{m} end{matrix} right] left[ begin{matrix} mathbf{x_1}(t)  mathbf{x_2}(t) end{matrix} right] + left[ begin{matrix} 0  frac{1}{m} end{matrix} right] mathbf{u}(t)
mathbf{y}(t) = left[ begin{matrix} 1 & 0 end{matrix} right] left[ begin{matrix} mathbf{x_1}(t)  mathbf{x_2}(t) end{matrix} right]

where

  • x1(t) represents the position of the object
  • x_2(t) := dot{x_1}(t) is the velocity of the object
  • dot{x_2}(t) = ddot{x_1}(t) is the acceleration of the object
  • the output mathbf{y}(t) is the position of the object

The controllability test is then Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. ...

left[ begin{matrix} B & AB end{matrix} right] = left[ begin{matrix} left[ begin{matrix} 0  frac{1}{m} end{matrix} right] & left[ begin{matrix} 0 & 1  -frac{k_2}{m} & -frac{k_1}{m} end{matrix} right] left[ begin{matrix} 0  frac{1}{m} end{matrix} right] end{matrix} right] = left[ begin{matrix} 0 & frac{1}{m}  frac{1}{m} & frac{k_1}{m^2} end{matrix} right]

which has full rank for all k1 and m.


The observability test is then Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. ...

left[ begin{matrix} C  CA end{matrix} right] = left[ begin{matrix} left[ begin{matrix} 1 & 0 end{matrix} right]  left[ begin{matrix} 1 & 0 end{matrix} right] left[ begin{matrix} 0 & 1  -frac{k_2}{m} & -frac{k_1}{m} end{matrix} right] end{matrix} right] = left[ begin{matrix} 1 & 0  0 & 1 end{matrix} right]

which also has full rank. Ergo, this system is both controllable and observable.


Nonlinear systems

The more general form of a state space model can be written as two functions.

mathbf{dot{x}}(t) = mathbf{f}(t, x(t), u(t))
mathbf{y}(t) = mathbf{h}(t, x(t), u(t))

The first is the state equation and the latter is the output equation. If the function f(cdot,cdot,cdot) is a linear combination of states and inputs then the equations can be written in matrix notation like above. The u(t) argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).


Pendulum example

A classic nonlinear system is a simple unforced pendulum Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ...

mlddottheta(t)= -mgsintheta(t) - kldottheta(t)

where

  • θ(t) is the angle of the pendulum with respect to the direction of gravity
  • m is the mass of the pendulum (pendulum rod's mass is assumed to be zero)
  • g is the gravitational acceleration
  • k is coefficient of friction at the pivot point
  • l is the radius of the pendulum (to the center of gravity of the mass m)

The state equations are then

where

  • x1(t): = θ(t) is the angle of the pendulum
  • is the rotational velocity of the pendulum
  • is the rotational acceleration of the pendulum

Instead, the state equation can be written in the general form

The equilibrium/stationary points of a system are when and so the equilibrium points of a pendulum are those that satisfy A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... Stationary points (red pluses) and inflection points (green circles). ...

for integers n.


References

  • Chen, Chi-Tsong 1999. Linear System Theory and Design, 3rd. ed., Oxford University Press (ISBN 0-19-511777-8)
  • Khalil, Hassan K. Nonlinear Systems, 3rd. ed., Prentice Hall (ISBN 0-13-067389-7)
  • Nise, Norman S. 2004. Control Systems Engineering, 4th ed., John Wiley & Sons, Inc. (ISBN 0-471-44577-0)
  • Hinrichsen, Diederich and Pritchard, Anthony J. 2005. Mathematical Systems Theory I, Modelling, State Space Analysis, Stability and Robustness. Springer. (ISBN 978-3-540-44125-0)
  • Sontag, Eduardo D. 1999. Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. (ISBN 0-387-984895) (available free online)

On the applications of state space models in econometrics:

  • Durbin, J. and S. Koopman (2001). Time series analysis by state space methods. Oxford University Press, Oxford.

See also

  • Phase space for information about phase state (like state space) in physics and mathematics.
  • State space for information about state space with discrete states in computer science.
  • State space (physics) for information about state space in physics.

  Results from FactBites:
 
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