State-space representation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/State-space_representation

In control engineering, model based fault detection and system identification a state-space representation is a mathematical model of a physical system specified as a set of input, output and variables related by first-order (not involving second derivatives) differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables.

The state space or phase space is the geometric space in which the variables on the axes are the state variables. The state of the system can be represented as a vector, the state vector, within state space.

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 ${\displaystyle p}$ inputs and ${\displaystyle q}$ outputs, we would otherwise have to write down ${\displaystyle q\times p}$ 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.

The state-space model can be applied in subjects such as economics, statistics, computer science and electrical engineering, and neuroscience. In econometrics, for example, state-space models can be used to decompose a time series into trend and cycle, compose individual indicators into a composite index, identify turning points of the business cycle, and estimate GDP using latent and unobserved time series. Many applications rely on the Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations.

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, ${\displaystyle n}$, is usually equal to the order of the system's defining differential equation, but not necessarily. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of 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 electric circuits, the number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as capacitors and inductors. The state variables defined must be linearly independent, i.e., no state variable can be written as a linear combination of the other state variables, or the system cannot be solved.

Linear systems

Block diagram representation of the linear state-space equations

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

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

${\displaystyle \mathbf{y}(t)=\mathbf{C}(t)\mathbf{x}(t)+\mathbf{D}(t)\mathbf{u}(t)}$

where:

${\displaystyle \mathbf{x}(\cdot )}$ is called the "state vector",  ${\displaystyle \mathbf{x}(t)\in {\mathbb{R}}^n}$;

${\displaystyle \mathbf{y}(\cdot )}$ is called the "output vector",  ${\displaystyle \mathbf{y}(t)\in {\mathbb{R}}^q}$;

${\displaystyle \mathbf{u}(\cdot )}$ is called the "input (or control) vector",  ${\displaystyle \mathbf{u}(t)\in {\mathbb{R}}^p}$;

${\displaystyle \mathbf{A}(\cdot )}$ is the "state (or system) matrix",  ${\displaystyle \dim [\mathbf{A}(\cdot )]=n\times n}$,

${\displaystyle \mathbf{B}(\cdot )}$ is the "input matrix",  ${\displaystyle \dim [\mathbf{B}(\cdot )]=n\times p}$,

${\displaystyle \mathbf{C}(\cdot )}$ is the "output matrix",  ${\displaystyle \dim [\mathbf{C}(\cdot )]=q\times n}$,

${\displaystyle \mathbf{D}(\cdot )}$ is the "feedthrough (or feedforward) matrix" (in cases where the system model does not have a direct feedthrough, ${\displaystyle \mathbf{D}(\cdot )}$ is the zero matrix),  ${\displaystyle \dim [\mathbf{D}(\cdot )]=q\times p}$,

${\displaystyle \dot{\mathbf{x}}(t):=\frac{d}{dt}\mathbf{x}(t)}$.

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 ${\displaystyle t}$ can be continuous (e.g. ${\displaystyle t\in \mathbb{R}}$) or discrete (e.g. ${\displaystyle t\in \mathbb{Z}}$). In the latter case, the time variable ${\displaystyle k}$ is usually used instead of ${\displaystyle t}$. Hybrid systems allow for time domains that have both continuous and discrete parts. Depending on the assumptions made, the state-space model representation can assume the following forms:

System type	State-space model
Continuous time-invariant	${\displaystyle \dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)}$ ${\displaystyle \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{u}(t)}$
Continuous time-variant	${\displaystyle \dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t)}$ ${\displaystyle \mathbf{y}(t)=\mathbf{C}(t)\mathbf{x}(t)+\mathbf{D}(t)\mathbf{u}(t)}$
Explicit discrete time-invariant	${\displaystyle \mathbf{x}(k+1)=\mathbf{A}\mathbf{x}(k)+\mathbf{B}\mathbf{u}(k)}$ ${\displaystyle \mathbf{y}(k)=\mathbf{C}\mathbf{x}(k)+\mathbf{D}\mathbf{u}(k)}$
Explicit discrete time-variant	${\displaystyle \mathbf{x}(k+1)=\mathbf{A}(k)\mathbf{x}(k)+\mathbf{B}(k)\mathbf{u}(k)}$ ${\displaystyle \mathbf{y}(k)=\mathbf{C}(k)\mathbf{x}(k)+\mathbf{D}(k)\mathbf{u}(k)}$
Laplace domain of continuous time-invariant	${\displaystyle s\mathbf{X}(s)-\mathbf{x}(0)=\mathbf{A}\mathbf{X}(s)+\mathbf{B}\mathbf{U}(s)}$ ${\displaystyle \mathbf{Y}(s)=\mathbf{C}\mathbf{X}(s)+\mathbf{D}\mathbf{U}(s)}$
Z-domain of discrete time-invariant	${\displaystyle z\mathbf{X}(z)-z\mathbf{x}(0)=\mathbf{A}\mathbf{X}(z)+\mathbf{B}\mathbf{U}(z)}$ ${\displaystyle \mathbf{Y}(z)=\mathbf{C}\mathbf{X}(z)+\mathbf{D}\mathbf{U}(z)}$

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 ${\displaystyle \mathbf{A}}$. 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:

${\displaystyle \mathbf{\text{G}}(s)=k\frac{(s-z_1)(s-z_2)(s-z_3)}{(s-p_1)(s-p_2)(s-p_3)(s-p_4)}.}$

The denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of ${\displaystyle s\mathbf{I}-\mathbf{A}}$,

${\displaystyle \lambda (s)=|s\mathbf{I}-\mathbf{A}|.}$

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 ${\displaystyle \mathbf{\text{G}}(s)}$ can similarly be used to determine whether the system is minimum phase.

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

Main article: Controllability

The 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

${\displaystyle \mathrm{rank}\left[\begin{array}{ccccc}\mathbf{B}& \mathbf{A}\mathbf{B}& {\mathbf{A}}^2\mathbf{B}& \cdots & {\mathbf{A}}^{n-1}\mathbf{B}\end{array}\right]=n,}$

where rank is the number of linearly independent rows in a matrix, and where n is the number of state variables.

Observability

Main article: 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

${\displaystyle \mathrm{rank}\left[\begin{array}{c}\mathbf{C}\\ \mathbf{C}\mathbf{A}\\ ⋮\\ \mathbf{C}{\mathbf{A}}^{n-1}\end{array}\right]=n.}$

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

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

yields

${\displaystyle s\mathbf{X}(s)-\mathbf{x}(0)=\mathbf{A}\mathbf{X}(s)+\mathbf{B}\mathbf{U}(s).}$

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

${\displaystyle (s\mathbf{I}-\mathbf{A})\mathbf{X}(s)=\mathbf{x}(0)+\mathbf{B}\mathbf{U}(s)}$

and thus

${\displaystyle \mathbf{X}(s)=(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{x}(0)+(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{B}\mathbf{U}(s).}$

Substituting for ${\displaystyle \mathbf{X}(s)}$ in the output equation

${\displaystyle \mathbf{Y}(s)=\mathbf{C}\mathbf{X}(s)+\mathbf{D}\mathbf{U}(s),}$

giving

${\displaystyle \mathbf{Y}(s)=\mathbf{C}((s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{x}(0)+(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{B}\mathbf{U}(s))+\mathbf{D}\mathbf{U}(s).}$

Assuming zero initial conditions ${\displaystyle \mathbf{x}(0)=\mathbf{0}}$ and a single-input single-output (SISO) system, the transfer function is defined as the ratio of output and input ${\displaystyle G(s)=Y(s)/U(s)}$. For a multiple-input multiple-output (MIMO) system, however, this ratio is not defined. Therefore, assuming zero initial conditions, the transfer function matrix is derived from

${\displaystyle \mathbf{Y}(s)=\mathbf{G}(s)\mathbf{U}(s)}$

using the method of equating the coefficients which yields

${\displaystyle \mathbf{G}(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{B}+\mathbf{D}}$.

Consequently, ${\displaystyle \mathbf{G}(s)}$ is a matrix with the dimension ${\displaystyle q\times p}$ which contains transfer functions for each input output combination. Due to the simplicity of this matrix notation, the state-space representation is commonly used for multiple-input, multiple-output systems. The Rosenbrock system matrix provides a bridge between the state-space representation and its transfer function.

Canonical realizations

Main article: Realization (systems)

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:

${\displaystyle \mathbf{\text{G}}(s)=\frac{n_1{s}^3+n_2{s}^2+n_{3}s+n_4}{s^4+d_1{s}^3+d_2{s}^2+d_{3}s+d_4}.}$

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

${\displaystyle \dot{\mathbf{x}}(t)=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ -d_4& -d_3& -d_2& -d_1\end{array}\right]\mathbf{x}(t)+\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\mathbf{u}(t)}$

${\displaystyle \mathbf{y}(t)=\left[\begin{array}{cccc}{n}_4& n_3& n_2& n_1\end{array}\right]\mathbf{x}(t).}$

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

${\displaystyle \dot{\mathbf{\text{x}}}(t)=\left[\begin{array}{cccc}0& 0& 0& -d_4\\ 1& 0& 0& -d_3\\ 0& 1& 0& -d_2\\ 0& 0& 1& -d_1\end{array}\right]\mathbf{\text{x}}(t)+\left[\begin{array}{c}{n}_4\\ n_3\\ n_2\\ n_1\end{array}\right]\mathbf{\text{u}}(t)}$

${\displaystyle \mathbf{\text{y}}(t)=\left[\begin{array}{cccc}0& 0& 0& 1\end{array}\right]\mathbf{\text{x}}(t).}$

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.

${\displaystyle \mathbf{\text{G}}(s)={\mathbf{\text{G}}}_{\mathrm{S}\mathrm{P}}(s)+\mathbf{\text{G}}(\mathrm{\infty }).}$

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 ${\displaystyle \mathbf{\text{y}}(t)=\mathbf{\text{G}}(\mathrm{\infty })\mathbf{\text{u}}(t)}$. 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:

${\displaystyle \mathbf{\text{G}}(s)=\frac{s^2+3s+3}{s^2+2s+1}=\frac{s+2}{s^2+2s+1}+1}$

which yields the following controllable realization

${\displaystyle \dot{\mathbf{\text{x}}}(t)=\left[\begin{array}{cc}-2& -1\\ 1& 0\end{array}\right]\mathbf{\text{x}}(t)+\left[\begin{array}{c}1\\ 0\end{array}\right]\mathbf{\text{u}}(t)}$

${\displaystyle \mathbf{\text{y}}(t)=\left[\begin{array}{cc}1& 2\end{array}\right]\mathbf{\text{x}}(t)+\left[\begin{array}{c}1\end{array}\right]\mathbf{\text{u}}(t)}$

Notice how the output also depends directly on the input. This is due to the ${\displaystyle \mathbf{\text{G}}(\mathrm{\infty })}$ constant in the transfer function.

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: ${\displaystyle \mathbf{u}(t)=K\mathbf{y}(t)}$. 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.

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

${\displaystyle \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)}$

becomes

${\displaystyle \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+BK\mathbf{y}(t)}$

${\displaystyle \mathbf{y}(t)=C\mathbf{x}(t)+DK\mathbf{y}(t)}$

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

${\displaystyle \dot{\mathbf{x}}(t)=\left(A+BK{\left(I-DK\right)}^{-1}C\right)\mathbf{x}(t)}$

${\displaystyle \mathbf{y}(t)={\left(I-DK\right)}^{-1}C\mathbf{x}(t)}$

The advantage of this is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of ${\displaystyle \left(A+BK{\left(I-DK\right)}^{-1}C\right)}$. This assumes that the closed-loop system is controllable or that the unstable eigenvalues of A can be made stable through appropriate choice of K.

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

${\displaystyle \dot{\mathbf{x}}(t)=\left(A+BK\right)\mathbf{x}(t)}$

${\displaystyle \mathbf{y}(t)=\mathbf{x}(t)}$

This reduces the necessary eigendecomposition to just ${\displaystyle A+BK}$.

Feedback with setpoint (reference) input

Output feedback with set point

In addition to feedback, an input, ${\displaystyle r(t)}$, can be added such that ${\displaystyle \mathbf{u}(t)=-K\mathbf{y}(t)+\mathbf{r}(t)}$.

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

${\displaystyle \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)}$

becomes

${\displaystyle \dot{\mathbf{x}}(t)=A\mathbf{x}(t)-BK\mathbf{y}(t)+B\mathbf{r}(t)}$

${\displaystyle \mathbf{y}(t)=C\mathbf{x}(t)-DK\mathbf{y}(t)+D\mathbf{r}(t)}$

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

${\displaystyle \dot{\mathbf{x}}(t)=\left(A-BK{\left(I+DK\right)}^{-1}C\right)\mathbf{x}(t)+B\left(I-K{\left(I+DK\right)}^{-1}D\right)\mathbf{r}(t)}$

${\displaystyle \mathbf{y}(t)={\left(I+DK\right)}^{-1}C\mathbf{x}(t)+{\left(I+DK\right)}^{-1}D\mathbf{r}(t)}$

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

${\displaystyle \dot{\mathbf{x}}(t)=\left(A-BKC\right)\mathbf{x}(t)+B\mathbf{r}(t)}$

${\displaystyle \mathbf{y}(t)=C\mathbf{x}(t)}$

Moving object example

A classical linear system is that of one-dimensional movement of an object (e.g., a cart). Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring:

${\displaystyle m\ddot{y}(t)=u(t)-b\dot{y}(t)-ky(t)}$

where

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

The state equation would then become

${\displaystyle \left[\begin{array}{c}{\dot{\mathbf{x}}}_1(t)\\ {\dot{\mathbf{x}}}_2(t)\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -\frac{k}{m}& -\frac{b}{m}\end{array}\right]\left[\begin{array}{c}{\mathbf{x}}_1(t)\\ {\mathbf{x}}_2(t)\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right]\mathbf{u}(t)}$

${\displaystyle \mathbf{y}(t)=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{\mathbf{x}}_{\mathbf{1}}(t)\\ {\mathbf{x}}_{\mathbf{2}}(t)\end{array}\right]}$

where

• ${\displaystyle x_1(t)}$ represents the position of the object
• ${\displaystyle x_2(t)={\dot{x}}_1(t)}$ is the velocity of the object
• ${\displaystyle {\dot{x}}_2(t)={\ddot{x}}_1(t)}$ is the acceleration of the object
• the output ${\displaystyle \mathbf{y}(t)}$ is the position of the object

The controllability test is then

${\displaystyle \left[\begin{array}{cc}B& AB\end{array}\right]=\left[\begin{array}{cc}\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right]& \left[\begin{array}{cc}0& 1\\ -\frac{k}{m}& -\frac{b}{m}\end{array}\right]\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right]\end{array}\right]=\left[\begin{array}{cc}0& \frac{1}{m}\\ \frac{1}{m}& -\frac{b}{m^2}\end{array}\right]}$

which has full rank for all ${\displaystyle b}$ and ${\displaystyle m}$. This means, that if initial state of the system is known (${\displaystyle y(t)}$, ${\displaystyle \dot{y}(t)}$, ${\displaystyle \ddot{y}(t)}$), and if the ${\displaystyle b}$ and ${\displaystyle m}$ are constants, then there is a force ${\displaystyle u}$ that could move the cart into any other position in the system.

The observability test is then

${\displaystyle \left[\begin{array}{c}C\\ CA\end{array}\right]=\left[\begin{array}{c}\left[\begin{array}{cc}1& 0\end{array}\right]\\ \left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ -\frac{k}{m}& -\frac{b}{m}\end{array}\right]\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}$

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.

${\displaystyle \dot{\mathbf{x}}(t)=\mathbf{f}(t,x(t),u(t))}$

${\displaystyle \mathbf{y}(t)=\mathbf{h}(t,x(t),u(t))}$

The first is the state equation and the latter is the output equation. If the function ${\displaystyle f(\cdot ,\cdot ,\cdot )}$ is a linear combination of states and inputs then the equations can be written in matrix notation like above. The ${\displaystyle u(t)}$ 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

${\displaystyle m{\ell }^2\ddot{\theta }(t)=-m\ell g\sin \theta (t)-k\ell \dot{\theta }(t)}$

where

• ${\displaystyle \theta (t)}$ is the angle of the pendulum with respect to the direction of gravity
• ${\displaystyle m}$ is the mass of the pendulum (pendulum rod's mass is assumed to be zero)
• ${\displaystyle g}$ is the gravitational acceleration
• ${\displaystyle k}$ is coefficient of friction at the pivot point
• ${\displaystyle \ell }$ is the radius of the pendulum (to the center of gravity of the mass ${\displaystyle m}$)

The state equations are then

${\displaystyle {\dot{x}}_1(t)=x_2(t)}$

${\displaystyle {\dot{x}}_2(t)=-\frac{g}{\ell }\sin x_1(t)-\frac{k}{m\ell }{x}_2(t)}$

where

• ${\displaystyle x_1(t)=\theta (t)}$ is the angle of the pendulum
• ${\displaystyle x_2(t)={\dot{x}}_1(t)}$ is the rotational velocity of the pendulum
• ${\displaystyle {\dot{x}}_2={\ddot{x}}_1}$ is the rotational acceleration of the pendulum

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

${\displaystyle \dot{\mathbf{x}}(t)=\left[\begin{array}{c}{\dot{x}}_1(t)\\ {\dot{x}}_2(t)\end{array}\right]=\mathbf{f}(t,x(t))=\left[\begin{array}{c}{x}_2(t)\\ -\frac{g}{\ell }\sin x_1(t)-\frac{k}{m\ell }{x}_2(t)\end{array}\right].}$

The equilibrium/stationary points of a system are when ${\displaystyle \dot{x}=0}$ and so the equilibrium points of a pendulum are those that satisfy

${\displaystyle \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}n\pi \\ 0\end{array}\right]}$

for integers n.

Earth's mantle

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Earth%27s_mantle

The internal structure of Earth

Earth's mantle is a layer of silicate rock between the crust and the outer core. It has a mass of 4.01×10²⁴ kg (8.84×10²⁴ lb) and thus makes up 67% of the mass of Earth. It has a thickness of 2,900 kilometers (1,800 mi) making up about 84% of Earth's volume. It is predominantly solid but, on geologic time scales, it behaves as a viscous fluid, sometimes described as having the consistency of caramel. Partial melting of the mantle at mid-ocean ridges produces oceanic crust, and partial melting of the mantle at subduction zones produces continental crust.

Structure

Rheology

Earth's mantle is divided into two major rheological layers: the rigid lithosphere comprising the uppermost mantle, and the more ductile asthenosphere, separated by the lithosphere-asthenosphere boundary. Lithosphere underlying ocean crust has a thickness of around 100 km (62 mi), whereas lithosphere underlying continental crust generally has a thickness of 150–200 km (93–124 mi). The lithosphere and overlying crust make up tectonic plates, which move over the asthenosphere.

The Earth's mantle is divided into three major layers defined by sudden changes in seismic velocity:

• the upper mantle (starting at the Moho, or base of the crust around 7 to 35 km [4.3 to 21.7 mi] downward to 410 km [250 mi])
• the transition zone (approximately 410–660 km [250–410 mi]), in which wadsleyite (≈ 410–520 km [250–320 mi]) and ringwoodite (≈ 525–660 km [326–410 mi]) are stable
• the lower mantle (approximately 660–2,891 km [410–1,796 mi]), in which bridgmanite (≈ 660–2,685 km [410–1,668 mi]) and post-perovskite (≈ 2,685–2,891 km [1,668–1,796 mi]) are stable

The lower ~200 km of the lower mantle constitutes the D" (D-double-prime) layer, a region with anomalous seismic properties. This region also contains large low-shear-velocity provinces and ultra low velocity zones.

Mineralogical structure

Mineral transformations in the mantle

The top of the mantle is defined by a sudden increase in seismic velocity, which was first noted by Andrija Mohorovičić in 1909; this boundary is now referred to as the Mohorovičić discontinuity or "Moho".

The upper mantle is dominantly peridotite, composed primarily of variable proportions of the minerals olivine, clinopyroxene, orthopyroxene, and an aluminous phase. The aluminous phase is plagioclase in the uppermost mantle, then spinel, and then garnet below ~100 km (62 mi). Gradually through the upper mantle, pyroxenes become less stable and transform into majoritic garnet.

At the top of the transition zone, olivine undergoes isochemical phase transitions to wadsleyite and ringwoodite. Unlike nominally anhydrous olivine, these high-pressure olivine polymorphs have a large capacity to store water in their crystal structure. This has led to the hypothesis that the transition zone may host a large quantity of water. At the base of the transition zone, ringwoodite decomposes into bridgmanite (formerly called magnesium silicate perovskite), and ferropericlase. Garnet also becomes unstable at or slightly below the base of the transition zone.

The lower mantle is composed primarily of bridgmanite and ferropericlase, with minor amounts of calcium perovskite, calcium-ferrite structured oxide, and stishovite. In the lowermost ~200 km (120 mi) of the mantle, bridgmanite isochemically transforms into post-perovskite.

Composition

Green xenoliths of peridotite from the mantle are surrounded by black volcanic lava. These peridotite xenoliths were carried upward from the mantle by molten magma during a volcanic eruption in Arizona.

The chemical composition of the mantle is difficult to determine with a high degree of certainty because it is largely inaccessible. Rare exposures of mantle rocks occur in ophiolites, where sections of oceanic lithosphere have been obducted onto a continent. Mantle rocks are also sampled as xenoliths within basalts or kimberlites.

Composition of the Earth's upper mantle (depleted MORB)

Compound	Mass percent
SiO2	44.71
MgO	38.73
FeO	8.18
Al2O3	3.98
CaO	3.17
Cr2O3	0.57
NiO	0.24
MnO	0.13
Na2O	0.13
TiO2	0.13
P2O5	0.019
K2O	0.006

Most estimates of the mantle composition are based on rocks that sample only the uppermost mantle. There is debate as to whether the rest of the mantle, especially the lower mantle, has the same bulk composition. The mantle's composition has changed through the Earth's history due to the extraction of magma that solidified to form oceanic crust and continental crust.

It has also been proposed in a 2018 study that an exotic form of water known as ice VII can form from supercritical water in the mantle when diamonds containing pressurized water bubbles move upward, cooling the water to the conditions needed for ice VII to form.

Temperature and pressure

In the mantle, temperatures range from approximately 500 K (227 °C; 440 °F) at the upper boundary with the crust to approximately 4,200 K (3,930 °C; 7,100 °F) at the core-mantle boundary. The temperature of the mantle increases rapidly in the thermal boundary layers at the top and bottom of the mantle, and increases gradually through the interior of the mantle. Although the higher temperatures far exceed the melting points of the mantle rocks at the surface (about 1,500 K [1,230 °C; 2,240 °F] for representative peridotite), the mantle is almost exclusively solid. The enormous lithostatic pressure exerted on the mantle prevents melting, because the temperature at which melting begins (the solidus) increases with pressure.

The pressure in the mantle increases from a few hundred megapascals at the Moho to 139 GPa (20,200,000 psi; 1,370,000 atm) at the core-mantle boundary.

Movement

Main article: Mantle convection

This figure is a snapshot of one time-step in a model of mantle convection. Colors closer to red are hot areas and colors closer to blue are cold areas. In this figure, heat received at the core–mantle boundary results in thermal expansion of the material at the bottom of the model, reducing its density and causing it to send plumes of hot material upwards. Likewise, cooling of material at the surface results in its sinking.

Because of the temperature difference between the Earth's surface and outer core and the ability of the crystalline rocks at high pressure and temperature to undergo slow, creeping, viscous-like deformation over millions of years, there is a convective material circulation in the mantle. Hot material upwells, while cooler (and heavier) material sinks downward. Downward motion of material occurs at convergent plate boundaries called subduction zones. Locations on the surface that lie over plumes are predicted to have high elevation (because of the buoyancy of the hotter, less-dense plume beneath) and to exhibit hot spot volcanism. The volcanism often attributed to deep mantle plumes is alternatively explained by passive extension of the crust, permitting magma to leak to the surface: the plate hypothesis.

The convection of the Earth's mantle is a chaotic process (in the sense of fluid dynamics), which is thought to be an integral part of the motion of plates. Plate motion should not be confused with continental drift which applies purely to the movement of the crustal components of the continents. The movements of the lithosphere and the underlying mantle are coupled since descending lithosphere is an essential component of convection in the mantle. The observed continental drift is a complicated relationship between the forces causing oceanic lithosphere to sink and the movements within Earth's mantle.

Although there is a tendency to larger viscosity at greater depth, this relation is far from linear and shows layers with dramatically decreased viscosity, in particular in the upper mantle and at the boundary with the core. The mantle within about 200 km (120 mi) above the core–mantle boundary appears to have distinctly different seismic properties than the mantle at slightly shallower depths; this unusual mantle region just above the core is called D″ ("D double-prime"), a nomenclature introduced over 50 years ago by the geophysicist Keith Bullen. D″ may consist of material from subducted slabs that descended and came to rest at the core–mantle boundary or from a new mineral polymorph discovered in perovskite called post-perovskite.

Earthquakes at shallow depths are a result of faulting; however, below about 50 km (31 mi) the hot, high pressure conditions ought to inhibit further seismicity. The mantle is considered to be viscous and incapable of brittle faulting. However, in subduction zones, earthquakes are observed down to 670 km (420 mi). A number of mechanisms have been proposed to explain this phenomenon, including dehydration, thermal runaway, and phase change. The geothermal gradient can be lowered where cool material from the surface sinks downward, increasing the strength of the surrounding mantle, and allowing earthquakes to occur down to a depth of 400 km (250 mi) and 670 km (420 mi).

The pressure at the bottom of the mantle is ~136 GPa (19,700,000 psi; 1,340,000 atm). Pressure increases as depth increases, since the material beneath has to support the weight of all the material above it. The entire mantle, however, is thought to deform like a fluid on long timescales, with permanent plastic deformation accommodated by the movement of point, line, and/or planar defects through the solid crystals composing the mantle. Estimates for the viscosity of the upper mantle range between 10¹⁹ and 10²⁴ Pa·s, depending on depth, temperature, composition, state of stress, and numerous other factors. Thus, the upper mantle can only flow very slowly. However, when large forces are applied to the uppermost mantle it can become weaker, and this effect is thought to be important in allowing the formation of tectonic plate boundaries.

Exploration

Exploration of the mantle is generally conducted at the seabed rather than on land because of the relative thinness of the oceanic crust as compared to the significantly thicker continental crust.

The first attempt at mantle exploration, known as Project Mohole, was abandoned in 1966 after repeated failures and cost over-runs. The deepest penetration was approximately 180 m (590 ft). In 2005 an oceanic borehole reached 1,416 metres (4,646 ft) below the sea floor from the ocean drilling vessel JOIDES Resolution.

More successful was the Deep Sea Drilling Project (DSDP) that operated from 1968 to 1983. Coordinated by the Scripps Institution of Oceanography at the University of California, San Diego, DSDP provided crucial data to support the seafloor spreading hypothesis and helped to prove the theory of plate tectonics. Glomar Challenger conducted the drilling operations. DSDP was the first of three international scientific ocean drilling programs that have operated over more than 40 years. Scientific planning was conducted under the auspices of the Joint Oceanographic Institutions for Deep Earth Sampling (JOIDES), whose advisory group consisted of 250 distinguished scientists from academic institutions, government agencies, and private industry from all over the world. The Ocean Drilling Program (ODP) continued exploration from 1985 to 2003 when it was replaced by the Integrated Ocean Drilling Program (IODP).

On 5 March 2007, a team of scientists on board the RRS James Cook embarked on a voyage to an area of the Atlantic seafloor where the mantle lies exposed without any crust covering, midway between the Cape Verde Islands and the Caribbean Sea. The exposed site lies approximately three kilometres beneath the ocean surface and covers thousands of square kilometres. A relatively difficult attempt to retrieve samples from the Earth's mantle was scheduled for later in 2007. The Chikyu Hakken mission attempted to use the Japanese vessel Chikyū to drill up to 7,000 m (23,000 ft) below the seabed. This is nearly three times as deep as preceding oceanic drillings.

A novel method of exploring the uppermost few hundred kilometres of the Earth was proposed in 2005, consisting of a small, dense, heat-generating probe which melts its way down through the crust and mantle while its position and progress are tracked by acoustic signals generated in the rocks. The probe consists of an outer sphere of tungsten about one metre in diameter with a cobalt-60 interior acting as a radioactive heat source. It was calculated that such a probe will reach the oceanic Moho in less than 6 months and attain minimum depths of well over 100 km (62 mi) in a few decades beneath both oceanic and continental lithosphere.

Exploration can also be aided through computer simulations of the evolution of the mantle. In 2009, a supercomputer application provided new insight into the distribution of mineral deposits, especially isotopes of iron, from when the mantle developed 4.5 billion years ago.

Earth's crust

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Earth%27s_crust 

Plates in the crust of Earth

Earth's crust is Earth's thick outer shell of rock, referring to less than 1% of Earth's radius and volume. It is the top component of the lithosphere, a division of Earth's layers that includes the crust and the upper part of the mantle. The lithosphere is broken into tectonic plates whose motion allows heat to escape the interior of the Earth into space.

The crust lies on top of the mantle, a configuration that is stable because the upper mantle is made of peridotite and is therefore significantly denser than the crust. The boundary between the crust and mantle is conventionally placed at the Mohorovičić discontinuity, a boundary defined by a contrast in seismic velocity.

Geologic provinces of the world (USGS)

Shield   Platform   Orogen   Basin   Large igneous province   Extended crust

Oceanic crust:   0–20 Ma   20–65 Ma   >65 Ma

The temperature of the crust increases with depth, reaching values typically in the range from about 100 °C (212 °F) to 600 °C (1,112 °F) at the boundary with the underlying mantle. The temperature increases by as much as 30 °C (54 °F) for every kilometer locally in the upper part of the crust.

Composition

Main articles: Abundance of elements in Earth's crust and Goldschmidt classification

Thickness of Earth's crust (km)

• 

  Abundance (atom fraction) of the chemical elements in Earth's upper continental crust as a function of the atomic number.
  The rarest elements in the crust (shown in yellow) are not the heaviest, but are rather the siderophile (iron-loving) elements in the Goldschmidt classification of elements. These have been depleted by being relocated deeper into Earth's core. Their abundance in meteoroid materials is higher. Additionally, tellurium and selenium have been depleted from the crust due to formation of volatile hydrides.

The crust of Earth is of two distinct types:

1. Oceanic: 5 km (3 mi) to 10 km (6 mi) thick and composed primarily of denser, more mafic rocks, such as basalt, diabase, and gabbro.
2. Continental: 30 km (20 mi) to 50 km (30 mi) thick and mostly composed of less dense, more felsic rocks, such as granite. In a few places, such as the Tibetan Plateau, the Altiplano, and the eastern Baltic Shield, the continental crust is thicker (50 km (30 mi) to 80 km (50 mi)).

The average thickness of the crust is about 15 km (9 mi) to 20 km (12 mi).

Because both the continental and oceanic crust are less dense than the mantle below, both types of crust "float" on the mantle. The surface of the continental crust is significantly higher than the surface of the oceanic crust, due to the greater buoyancy of the thicker, less dense continental crust (an example of isostasy). As a result, the continents form high ground surrounded by deep ocean basins.

The continental crust has an average composition similar to that of andesite, though the composition is not uniform, with the upper crust averaging a more felsic composition similar to that of dacite, while the lower crust averages a more mafic composition resembling basalt. The most abundant minerals in Earth's continental crust are feldspars, which make up about 41% of the crust by weight, followed by quartz at 12%, and pyroxenes at 11%.

Most Abundant Elements of Earth's Crust	Approximate % by weight	Oxide	Approximate % oxide by weight
O	46.6
Si	27.7	SiO2	60.6
Al	8.1	Al2O3	15.9
Fe	5.0	Fe as FeO	6.7
Ca	3.7	CaO	6.4
Na	2.7	Na2O	3.1
K	2.6	K2O	1.8
Mg	1.5	MgO	4.7
Ti	0.44	TiO2	0.7
P	0.10	P2O5	0.1

All the other constituents except water occur only in very small quantities and total less than 1%.

Continental crust is enriched in incompatible elements compared to the basaltic ocean crust and much enriched compared to the underlying mantle. The most incompatible elements are enriched by a factor of 50 to 100 in the continental crust relative to primitive mantle rock, while oceanic crust is enriched with incompatible elements by a factor of about 10.

The estimated average density of the continental crust is 2.835 g/cm³, with density increasing with depth from an average of 2.66 g/cm³ in the uppermost crust to 3.1 g/cm³ at the base of the crust.

In contrast to the continental crust, the oceanic crust is composed predominantly of pillow lava and sheeted dikes with the composition of mid-ocean ridge basalt, with a thin upper layer of sediments and a lower layer of gabbro.

Formation and evolution

Main article: Earth's crustal evolution

Further information: Formation of the Earth

Earth formed approximately 4.6 billion years ago from a disk of dust and gas orbiting the newly formed Sun. It formed via accretion, where planetesimals and other smaller rocky bodies collided and stuck, gradually growing into a planet. This process generated an enormous amount of heat, which caused early Earth to melt completely. As planetary accretion slowed, Earth began to cool, forming its first crust, called a primary or primordial crust. This crust was likely repeatedly destroyed by large impacts, then reformed from the magma ocean left by the impact. None of Earth's primary crust has survived to today; all was destroyed by erosion, impacts, and plate tectonics over the past several billion years.

Since then, Earth has been forming a secondary and tertiary crust, which correspond to oceanic and continental crust, respectively. Secondary crust forms at mid-ocean spreading centers, where partial-melting of the underlying mantle yields basaltic magmas and new ocean crust forms. This "ridge push" is one of the driving forces of plate tectonics, and it is constantly creating new ocean crust. Consequently, old crust must be destroyed, so opposite a spreading center, there is usually a subduction zone: a trench where an ocean plate is sinking back into the mantle. This constant process of creating a new ocean crust and destroying the old ocean crust means that the oldest ocean crust on Earth today is only about 200 million years old.

In contrast, the bulk of the continental crust is much older. The oldest continental crustal rocks on Earth have ages in the range from about 3.7 to 4.28 billion years  and have been found in the Narryer Gneiss Terrane in Western Australia, in the Acasta Gneiss in the Northwest Territories on the Canadian Shield, and on other cratonic regions such as those on the Fennoscandian Shield. Some zircon with age as great as 4.3 billion years has been found in the Narryer Gneiss Terrane. Continental crust is a tertiary crust, formed at subduction zones through recycling of subducted secondary (oceanic) crust.

The average age of Earth's current continental crust has been estimated to be about 2.0 billion years. Most crustal rocks formed before 2.5 billion years ago are located in cratons. Such an old continental crust and the underlying mantle asthenosphere are less dense than elsewhere on Earth and so are not readily destroyed by subduction. Formation of new continental crust is linked to periods of intense orogeny, which coincide with the formation of the supercontinents such as Rodinia, Pangaea and Gondwana. The crust forms in part by aggregation of island arcs including granite and metamorphic fold belts, and it is preserved in part by depletion of the underlying mantle to form buoyant lithospheric mantle. Crustal movement on continents may result in earthquakes, while movement under the seabed can lead to tidal waves.

Health literacy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Health_literacy 

The ability to read and understand medication instructions is a form of health literacy.

Health literacy is the ability to obtain, read, understand, and use healthcare information in order to make appropriate health decisions and follow instructions for treatment. There are multiple definitions of health literacy, in part, because health literacy involves both the context (or setting) in which health literacy demands are made (e.g., health care, media, internet or fitness facility) and the skills that people bring to that situation.

Since health literacy is a primary contributing factor to health disparities, it is a continued and increasing concern for health professionals. The 2003 National Assessment of Adult Literacy (NAAL) conducted by the US Department of Education found that 36% of participants scored as either "basic" or "below basic" in terms of their health literacy and concluded that approximately 80 million Americans have limited health literacy. These individuals have difficulty with common health tasks including reading the label of a prescribed drug. Several factors may influence health literacy. However, the following factors have been shown to strongly increase this risk: age (especially patients 65 years and older), limited English language proficiency or English as a second language, chronic conditions, less education, and lower socioeconomic status. Patients with low health literacy understand less about their medical conditions and treatments and overall report worse health status.

Various interventions, such as simplifying information and illustrations, avoiding jargon, using "teach-back" methods, and encouraging patients' questions, have improved health behaviors in persons with low health literacy. The proportion of adults aged 18 and over in the U.S., in the year 2010, who reported that their health care providers always explained things so they could understand them was about 60.6%. This number increased 1% from 2007 to 2010. The Healthy People 2020 initiative of the United States Department of Health and Human Services (HHS) has included health literacy as a pressing new topic, with objectives for improving it in the decade to come.

In planning for Healthy People 2030 (the fifth edition of Healthy People), HHS issued a "Solicitation for Written Comments on an Updated Health Literacy Definition for Healthy People." Several proposals address the fact that "health literacy is multidimensional", being the result of a concerted effort that involves the individual seeking care or information, providers and caregivers, the complexity and demands of the system, and the use of plain language for communication.

Characteristics

A poster about airborne disease transmission encouraging the use of a handkerchief to prevent the spread.

Plain language

Plain Language refers to the use of writing strategies that help readers find, understand, and apply information to fulfill their needs. It has a vital role to play in improving health literacy. In conjunction with readers education, provider cultural training, and system design, plain language helps people make more informed health choices.

Plain language is not, however, a one-size-fits-all approach. Some strategies can be used to improve communication among medical professionals, while others will improve communication with patients and caregivers. It is in this area of provider-patient communication that health literacy efforts have been strongest. Patients and consumers in general need to understand concepts that professionals often refer to in technical terms. Health professionals must know their audience in order to better serve their patients and general readers or listeners.

Plain language is defined by the International Plain Language Federation as writing whose "wording, structure, and design are so clear that the intended readers can easily find what they need, understand what they find, and use that information."

Some key elements of plain language include:

• Organizing information so most important points come first
• Breaking complex information into understandable chunks
• Using simple language or language familiar to the reader
• Defining technical terms and acronyms
• Using active voice in subject-verb-object (SVO ) sentences when subject/agent/topic coincide
• Varying sentence length and structure to avoid monotony
• Using lists and tables to make complex material easier to understand

The National Institute of Health (NIH) recommends that patient education materials be written at a 6th–7th grade reading level; further recommendations provided by the NIH Office of Communications and Public Liaison are published in their "Clear Communication" Initiative. The UK's National Institute for Health and Care Research (NIHR) recommends involving patients and non-academic members of the public in writing plain language summaries of research articles.

Factors that contribute to health literacy

Many factors determine the health literacy level of health education materials or interventions: readability of the text, the patient's current state of health, language barriers between the clinician and patient, cultural appropriateness of the materials, format and style, sentence structure, use of illustrations, and numerous other factors.

A study of 69,000 patients conducted in 1995 by two US hospitals found that between 26% and 60% of patients could not understand medication directions, a standard informed consent form, or materials about scheduling an appointment. The 2003 National Assessment of Adult Literacy (NAAL) conducted by the US Department of Education found that 36% of participants scored as either "basic" or "below basic" in terms of their health literacy and concluded that approximately 80 million Americans have limited health literacy.

Results of a systematic review of the literature found that when limited English proficient (LEP) patients receive care from physicians who are fluent in the patients' preferred language, referred to as having language concordance, generally improves outcomes. These outcomes are consistent across patient-reported measures, such as patient satisfaction, and also more such as blood pressure for patients with diabetes.

Standardized measures of health literacy are the Newest Vital Sign (NVS), which asks people about a nutrition label, and the Test of Functional Health Literacy (TOFHLA), which asks test-takers to fill in 36 blanks in patient instructions for X-rays and a Medicaid application, from multiple choices, and 4 numbers in medicine dosage forms.

History

The young and multidisciplinary field of health literacy emerged from two groups of experts: physicians, health providers such as nurses and health educators; and Adult Basic Education (ABE) and English as a second language (ESL) practitioners in the field of education. Physicians and nurses are a source of patient comprehension and compliance studies. Adult Basic Education / English for Speakers of Languages Other Than English (ABE/ESOL) specialists study and design interventions to help people develop reading, writing, and conversation skills and increasingly infuse curricula with health information to promote better health literacy. A range of approaches to adult education brings health literacy skills to people in traditional classroom settings, as well as where they work and live.

Biomedical approach

The biomedical approach to health literacy that became dominant (in the U.S.) during the 1980s and 1990s often depicted individuals as lacking health literacy or "suffering" from low health literacy. This approach assumed that recipients are passive in their possession and reception of health literacy and believed that models of literacy and health literacy are politically neutral and universally applicable. This approach is found lacking when placed in the context of broader ecological, critical, and cultural approaches to health. This approach has produced, and continues to reproduce, numerous correlational studies.

Level of health literacy is considered adequate when the population has sufficient knowledge, skills, and confidence to guide their own health, and people are able to stay healthy, recover from illness, and/or live with disability or disease.

McMurray states that health literacy ist important in a community because it addresses health inequities. It is no coincidence that individuals with lower levels of health literacy live, disproportionately, in communities with lower socio-economic standing. A barrier to achieving adequate health literacy for these individuals is a lack of awareness, or understanding of, information and resources relevant to improving their health. This knowledge gap arises from both patients being unable to understand information presented to them and hospitals' inadequate efforts and materials to address these literacy gaps.

A more robust view of health literacy includes the ability to understand scientific concepts, content, and health research; skills in spoken, written, and online communication; critical interpretation of mass media messages; navigating complex systems of health care and governance; knowledge and use of community capital and resources; and using cultural and indigenous knowledge in health decision making. This integrative view sees health literacy as a social determinant of health that offers a powerful opportunity to reduce inequities in health.

This perspective defines health literacy as the wide range of skills, and competencies that people develop over their lifetimes to seek out, comprehend, evaluate, and use health information and concepts to make informed choices, reduce health risks, and increase quality of life. While various definitions vary in wording, they all fall within this conceptual framework.

Defining health literacy in that manner builds the foundation for a multi-dimensional model of health literacy built around four central domains:

• fundamental literacy,
• scientific literacy,
• civic literacy, and
• cultural literacy.

The cited paper by Zarcadoolas et al. focussed on "the public discourse about terrorism and bioterrorism that dominated the mass media during the anthrax threat in the United States during 2001."

There are several tests, which have verified reliability in the academic literature that can be administered in order to test one's health literacy. Some of these tests include the Medical Term Recognition Test (METER), which was developed in the United States (2 minute administration time) for the clinical setting. The METER includes many words from the Rapid Estimate of Adult Literacy in Medicine (REALM) test. The Short Assessment of Health Literacy in Spanish and English populations (SAHL-S&E) uses word recognition and multiple choice questions to test a person's comprehension. The CHC-Test measures Critical Health Competencies and consists of 72 items designed to test a person's understanding of medical concepts, literature searching, basic statistics, and design of experiments and samples.

Patient safety and outcomes

According to an Institute of Medicine (2004) report, low health literacy negatively affects the treatment outcome and safety of care delivery. The lack of health literacy affects all segments of the population. However, it is disproportionate in certain demographic groups, such as the elderly, ethnic minorities, recent immigrants, individuals facing homelessness, and persons with low general literacy. These populations have a higher risk of hospitalization, longer hospital stays, are less likely to comply with treatment, are more likely to make errors with medication, and are more ill when they initially seek medical care.

The mismatch between a clinician's communication of content and a patient's ability to understand that content can lead to medication errors and adverse medical outcomes. Health literacy skills are not only a problem in the general population. Health care professionals (doctors, nurses, public health workers) can also have poor health literacy skills, such as a reduced ability to clearly explain health issues to patients and the public. In addition to tailoring the content of what health professionals communicate to their patients, a well arranged layout, pertinent illustrations, and intuitive format of written materials can improve the usability of health care literature. This in turn can help in effective communication between healthcare providers and their patients.

Outcomes of low levels of health literacy also include relative expenditures on health services. Because individuals with low health literacy are more likely to have adverse health statuses, their use of health services is also increased. This trend is compounded by other risk factors of low health literacy, including poverty. Homelessness and housing insecurity can hinder good health and recovery in attempts to better health circumstances, causing the exacerbation of poor health conditions. In these cases, a variety of health services may be used repeatedly as health issues are prolonged. Thus overall expenditures on health services is greater among populations with low health literacy and poor health. These costs may be left to individuals and families to pay which may further burden health conditions, or the costs may be left to a variety of institutions which in turn has broader implications for government funding and health care systems.

A review of studies that focused on health literacy and its associated costs concluded that low levels of health literacy is responsible for 3–5% of healthcare cost—approximately $143 to 7,798 per individual within the healthcare system. For example, studies have shown that the increased prevalence of poor health and low health literacy has resulted in a greater use of emergency services by homeless individuals. A study conducted in San Francisco showed that "72% of the total cost of emergency services may be attributed to the top 13% of homeless users". In this way, low health literacy produces financial outcomes as well as those of health.

Risk identification

Identifying a patient as having low health literacy is essential for a healthcare professional to conform their health intervention in a way that the patient will understand. When patients with low health literacy receive care that is tailored to their more limited medical knowledge base, results have shown that health behaviors drastically improve. This has been seen with: correct medication use and dosage, utilizing health screenings, as well as increased exercise and smoking cessation. Effective visual aids have shown to help supplement the information communicated by the doctor in the office. In particular, easily readable brochures and videos have shown to be very effective. Healthcare professionals can use many methods to attain patients' health literacy. A multitude of tests used during research studies and three minute assessments commonly used in doctors offices are examples of the variety of tests healthcare professionals can use to better understand their patients' health literacy.

The American Medical Association showed that asking simple single-item questions, such as "How confident are you in filling out medical forms by yourself?", is a very effective and direct way to understand from a patient's point of view how they feel about interacting with their healthcare provider and understanding their health condition.

Homelessness

Individuals facing homelessness constitute a population that holds intersectional identities, is highly mobile, and is often out of the public eye. Thus the difficulty of conducting research on this group has resulted in little information regarding homelessness as a condition that has increased risk of low health literacy levels among individuals. Nonetheless, studies that do exist indicate that homeless individuals experience increased prevalence of low health literacy and poor health—both physical and mental—due to vulnerabilities brought on by the insecurity of basic needs among homeless individuals. The combination of poor health and homelessness has been found to increase the risk for further decline in health status and increased housing insecurity, all of which is highly affected—and in many cases perpetuated—by low levels of health literacy.

Intervention

In order to be understood by patients with insufficient health literacy, health professionals must intervene to provide clear and concise information that can be more easily understood. Avoidance of medical jargon, illustrations of important concepts, and confirming information by a "teach back" method have shown to be effective tools to communicating essential health topics with health illiterate patients. A program called "Ask Me 3" is designed to bring public and physician attention to this issue, by letting patients know that they should ask three questions each time they talk to a doctor, nurse, or pharmacist:

• What is my main problem?
• What do I need to do?
• Why is it important for me to do this?

There have also been large-scale efforts to improve health literacy. For example, a public information program by the US Department of Health and Human Services encourages patients to improve healthcare quality and avoid errors by asking questions about health conditions and treatment. Additionally, the IROHLA (Intervention Research on Health Literacy of the Ageing population) project, funded by the European Union (EU), seeks to develop evidence-based guidelines for policy and practice to improve health literacy of the ageing population in EU member states. The project has developed a framework and identified and validated interventions which together constitute a comprehensive approach of addressing health literacy needs of the elderly. The UK's National Institute for Health and Care Research (NIHR) has published review of research on health literacy, focusing on how to help people understand health information, and which groups of the population may need extra support. It covers research on using plain language, finding the focus, online information, and using alternative formats (e.g. visuals).

Health illiterate-related diabetes prevalence in Vietnamese-American populations

Diabetes is a rapidly growing health problem among immigrants—affecting approximately 10 percent of Asian-Americans. It is the fifth-leading cause of death in Asian-Americans between the ages of 45 and 64. In addition, type 2 diabetes is the most common form of the disease. Those who are diagnosed with type 2 diabetes have high levels of blood glucose because the body does not effectively respond to insulin. It is a lifelong disease with no known cure. Diabetes is a chronic, debilitating, and costly social burden—costing healthcare systems about $100 billion annually.

Diabetes disproportionately affects underserved and ethnically diverse populations, such as Vietnamese-American communities. The relationship between the disease and health literacy level is in part because of an individual's ability to read English, evaluate blood glucose levels, and communicate with medical professionals. Other studies also suggest lack in knowledge of diabetes symptoms and complications. According to an observational cross-sectional study conducted, many Vietnamese-American diabetic patients show signs of poor blood glucose control and adherence due to inadequate self-management knowledge and experience. Diabetes health literacy research is needed to fully understand the burden of the chronic disease in Vietnamese-American communities, with respect to language and culture, health literacy, and immigrant status. Ethnic minority groups and immigrant communities have less knowledge of health promoting behavior, face considerable obstacles to health services, and experience poor communication with medical professionals. According to a recent review, studies have supported an independent relationship between literacy and knowledge of diabetes management and glucose control, but its impact on patients has not been sufficiently described. With the demand of chronic disease self-management (e.g., diabetic diet, glucose monitoring, etc.), a call for cultural-specific patient education is needed to achieve the control of diabetes and its adverse health outcomes in low- to middle-income Vietnamese-American immigrant communities.

Oral health literacy in school teachers of Mangalore, India

The problem of low oral health literacy (OHL) is often neglected which may lead to poor oral health outcomes and under utilization of oral care services. A cross-sectional survey of school teachers working in schools in Mangalore, India, was undertaken. Details regarding demographics, medical, and dental history, oral hygiene practices and habits, diet history, and decay promoting the potential of school teachers were obtained using face-to-face interview method. The Rapid Estimate of Adult Literacy in Dentistry-99 (REALD-99) was used to assess their OHL. The OHL was high in the school teachers with the REALD-99 scores ranging from 45 to 95 with a mean score of 75.83 ± 9.94. Th This study found that there was a statistically significant difference between OHL and education, frequency of brushing and the filled teeth. Although this study indicated high OHL levels among school teachers in Mangalore, India, the magnitude of dental caries in this population was also relatively high and very few had a healthy periodontium.

eHealth literacy

eHealth literacy describes an individual's ability to search for, access, comprehend, and appraise desired health information from electronic sources and to then use such information to attempt to address a particular health problem. It has become an important topic of research due to the increasing use of the internet for health information seeking and health information distribution. Stellefson (2011) states, "8 out of 10 Internet users report that they have at least once looked online for health information, making it the third most popular Web activity next to checking email and using search engines in terms of activities that almost everybody has done." Though in recent years, individuals may have gained access to a multitude of health information via the Internet, access alone does not ensure that proper search skills and techniques are being used to find the most relevant online and electronic resources. As the line between a reputable medical source and an amateur opinion can often be blurred, the ability to differentiate between the two is important.

Health literacy requires a combination of several different literacy skills in order to facilitate eHealth promotion and care. Six core skills are delineated by an eHealth literacy model referred to as the Lily model. The Lily Model's six literacies are organized into two central types: analytic and context-specific. Analytic type literacies are those skills that can be applied to a broad range of sources, regardless of topic or content (i.e., skills that can also be applied to shopping or researching a term paper in addition to health) whereas context-specific skills are those that are contextualized within a specific problem domain (can solely be applied to health). The six literacies are listed below, the first three of the analytic type and the latter three of the context-specific:

• Traditional literacy
• Media literacy
• Information literacy
• Computer literacy
• Scientific literacy
• Health literacy

According to Norman (2006), both analytical and context-specific literacy skills are "required to fully engage with electronic health resources." As the World Wide Web and technological innovations are more and more becoming a part of the healthcare environment, it is important for information technology to be properly utilized to promote health and deliver health care effectively.

It has also been suggested that the move towards patient-centered care and the greater use of technology for self-care and self-management requires higher health literacy on the part of the patient. This has been noted in several research studies, for example among adolescent patients with obesity.

Improvement

Incorporate information through the university level

The United States Department of Health and Human Services created a National Action Plan to Improve Health Literacy. One of the goals of the National Action Plan is to incorporate health and science information in childcare and education through the university level. The target is to educate people at an early stage; that way individuals are raised with health literacy and will have a better quality of life. The earlier an individual is exposed to health literacy skills the better for the person and the community.

Programs such as Head Start and Women, Infants, and Children (WIC) have impacted society, especially the low-income population. Head Start provides low-income children and their families early childhood education, nutrition, and health screenings. Health literacy is integrated in the program for both children and parents through the education given to the individuals. WIC serves low-income pregnant women and new mothers by supplying them with food, health care referrals, and nutrition education. Programs like these help improve the health literacy of both the parent and the child, creating a more knowledgeable community with health education.

Although programs like Head Start and WIC have been working with the health literacy of a specific population, much more can be done with the education of children and young adults. Now, more and more adolescents are getting involved with their health care. It is crucial to educate these individuals in order for them to make informed decisions.

Many schools in the country incorporate a health class in their curriculum. These classes provided an excellent opportunity to facilitate and develop health literacy in today's children and adolescents. The skills of how to read food labels, the meaning of common medical terms, the structure of the human body, and education on the most prevalent diseases in the United States should be taught in both private and public schools. This way new generations will grow with health literacy and would hopefully make knowledgeable health decisions.

Framework and potential intervention points

The National Library of Medicine defines health literacy as:

"The degree to which individuals have the capacity to obtain, process, and understand basic health literacy information and services needed to make appropriate health decisions."

Based on this clinical definition, health literacy gives individuals the skills that they need to both understand and effectively communicate information and concerns. Bridging that gap between literacy skills and the ability of the individual in health contexts, the Health Literacy Framework highlights the health outcomes and costs associated with health contexts including cognitive abilities, social skills, emotional state, and physical conditions such as visual and auditory contributions.

Potential Intervention Points are illustrated in reflection of the Health Literacy Framework. While these potential intervention points include interactions such as those of individuals and the education systems that they are engaged with, their health systems, and societal factors as they relate to health literacy, these points are not components of a causal model. The three potential intervention points are culture and society, the health system, and the education system. Health outcomes and costs are the products of the health literacy developed during diversity of exposure to these three potential intervention points.

Referring to shared ideas, meanings, and values that influence an individual's beliefs and attitudes, cultural and societal influences are a significant intervention point for health literacy development. As interactions with healthcare systems often first occur at the family level, deeply rooted beliefs and values can shape the significance of the experience. Included components that reflect the development of health literacy both culturally and societally are native language, socioeconomic status, gender, race, and ethnicity, as well as mass media exposure. These are pathways to understanding American life paralleling conquests for a health-literate America.

The health system is an intervention point in the Health Literacy Framework. For the purposes of this framework, health literacy refers to an individual's interaction with people performing health-related activities in settings such as hospitals, clinics, physician's offices, home health care, public health agencies, and insurers.

In the United States, the education system consists of K–12 curricula. In addition to this standard educational setting, adult education programs are also environments in which individuals can develop traditional literacy skills founded in comprehension and real-world application of knowledge via reading and writing. Tools for educational development provided by these systems impact an individual's capacity to obtain specific knowledge regarding health. Reflecting components of traditional literacy such as cultural and conceptual knowledge, oral literacy (listening and speaking,) print literacy (reading and writing,) and numeracy, education systems are also potential intervention points for health literacy development.

Development of a health literacy program

A successful health literacy program will have many goals that all work together to improve health literacy. Many people assume these goals should communicate health information to the general public; however, in order to be successful, the goals should not only communicate with people but also take into account social and environmental factors that influence lifestyle choices. A good example of this is the movement to end smoking. When a health literacy program is put into place where only the negative side effects of smoking are told to the general public it is doomed to fail. However, when there is a larger program put in – one that includes strategies outlining how to quit smoking, raises tobacco prices, reduces access to tobacco by minors, and reflects the social unacceptability of smoking – it will be much more effective.

The U.S. Department of Health and Human Services suggests a National Action Plan to implement a comprehensive Health Literacy Program. They include seven goals:

1. Develop and disseminate health and safety information that is accurate, accessible, and actionable
2. Promote changes in the health care system that improve health information, communication, informed decision making, and access to health services
3. Incorporate accurate, standards-based, and developmentally appropriate health and science information and curricula in child care and education through the university level
4. Support and expand local efforts to provide adult education, English language instruction, and culturally and linguistically appropriate health information services in the community
5. Build partnerships, develop guidance, and change policies
6. Increase basic research and the development, implementation, and evaluation of practices and interventions to improve health literacy
7. Increase the dissemination and use of evidence-based health literacy practices and interventions

These goals should be taken into account when implementing a health literacy program.

There are also goals for the outcomes of a Health Literacy Program.

Health-related goals:

• Promoting and protect health and prevent disease
• Understand, interpret, and analyze health information
• Apply health information over a variety of life events and situations
• Navigate the healthcare system
• Actively participate in encounters with healthcare professionals and workers
• Understand and give consent
• Understand and advocate for rights

In the creation of a program aimed to improve health literacy, it is also important to ensure that all parties involved in health contexts are on the same page. To do this, programs may choose to include the training of case managers, health advocates, and even doctors and nurses. Due to the common overestimations of health literacy levels of patients, the education of health literacy topics and training in the identification of low health literacy in patients may be able to create significant positive change in the understanding of health messages. The Health Belief Model has been used in the training of health professionals in order to share insight on the knowledge that it has been shown to most likely change health perceptions and behaviors of their patients. The use of the health belief model can provide basis for which patient health literacy may grow. The training of health workers may be seen as a "work around intervention" but is still a viable option and opportunity for mediating the negative outcomes of low health literacy. Effective health literacy programs are created with cultural competency, and individuals working within health institutions can support individuals with low health literacy by being culturally competent themselves.

In working to improve the health literacy of individuals, a multitude of approaches may be taken. Systematic reviews of studied interventions reveal that one works to improve health literacy in one patient may not work for another patient. In fact, some interventions were found to worse health literacy in individuals. Nonetheless, studies have illuminated general approaches that help individuals understand health messages. A review of 26 studies concluded that "intensive mixed-strategy interventions focusing on self-management" and "theory basis, pilot testing, emphasis on skill building, and delivery by a health professional" do aid in increasing levels of health literacy among patients. Another study revealed that programs aimed at targeting more than one behavior through increased health literacy are no less successful than programs with a single focus. The importance of dignity and respect is emphasized when creating programs for increasing health literacy of vulnerable individuals. In intervention programs created for homeless individuals in specific, it has been found that "successful intervention programs use aggressive outreach to bring comprehensive social and health services to sites where homeless people congregate and allow clients to set the limits and pace of engagement". A social justice model is recommended for homeless individuals which is based on shared support of the community and their health literacy needs by those who provide services for this underserved group as well as the professionals who create and implement health literacy interventions.

Role of libraries in health literacy

Libraries have increasingly recognized that they can play a role in health literacy since the 2000s, influenced by the Medical Library Association. Library initiatives have included running education programs, fostering partnerships with health organizations, and using outreach efforts. Pan American Health Organization contributed greatly through their libraries network. Various medical library associations have made an effort to introduce health literacy programs by attempting to define the concept to include the librarian's role as including a set of abilities needed to recognize a health information need, identify likely information sources and use them to retrieve relevant information, and assess the quality of the information and its applicability to a specific health situation. The Network of the National Library of Medicine (NNLM) twice yearly hosts the #citeNLM Wikipedia Edit-a-thon initiative to "improve the credibility and content of medical and health-related articles on Wikipedia by adding citations and information from National Library of Medicine (NLM) sources."

Alternative approaches

A study published in the American Journal of Obstetrics and Gynecology in 2018 suggested an alternative method of addressing health literacy. This alternative method is the use of "culturally-related/accepted" music and singing amongst low-literacy populations. In the example of this study, the music was shown to increase health literacy with regards to maternal health, and more specifically, antenatal care.