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In control engineering 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 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 state variable values and may also depend on the input variable values.
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector, the state vector.
If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. [1] [2] The state-space method is characterized by the 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 or without modulation. [3] 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 inputs and outputs, we would otherwise have to write down 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, [4] statistics, [5] computer science and electrical engineering, [6] and neuroscience. [7] 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, [8] identify turning points of the business cycle, and estimate GDP using latent and unobserved time series. [9] [10] Many applications rely on the Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations. [11] [12]
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. [13] The minimum number of state variables required to represent a given system, , 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.
The most general state-space representation of a linear system with inputs, outputs and state variables is written in the following form: [14]
where:
In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in the common LTI case, matrices will be time invariant. The time variable can be continuous (e.g. ) or discrete (e.g. ). In the latter case, the time variable is usually used instead of . 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 | |
Continuous time-variant | |
Explicit discrete time-invariant | |
Explicit discrete time-variant | |
Laplace domain of continuous time-invariant | |
Z-domain of discrete time-invariant |
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 . The stability of a time-invariant state-space model can be determined by looking at the system's transfer function in factored form. It will then look something like this:
The denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of ,
The roots of this polynomial (the eigenvalues) are the system transfer function's poles (i.e., the singularities where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is asymptotically stable or marginally stable. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability.
The zeros found in the numerator of 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).
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
where rank is the number of linearly independent rows in a matrix, and where n is the number of state variables.
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
The "transfer function" of a continuous time-invariant linear state-space model can be derived in the following way:
First, taking the Laplace transform of
yields
Next, we simplify for , giving
and thus
Substituting for in the output equation
giving
Assuming zero initial conditions and a single-input single-output (SISO) system, the transfer function is defined as the ratio of output and input . 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
using the method of equating the coefficients which yields
Consequently, is a matrix with the dimension 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.
Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system):
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
The coefficients can now be inserted directly into the state-space model by the following approach:
This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
The transfer function coefficients can also be used to construct another type of canonical form
This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
Transfer functions which are only proper (and not strictly proper) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant.
The strictly proper transfer function can then be transformed into a canonical state-space realization using techniques shown above. The state-space realization of the constant is trivially . 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:
which yields the following controllable realization
Notice how the output also depends directly on the input. This is due to the constant in the transfer function.
A common method for feedback is to multiply the output by a matrix K and setting this as the input to the system: . 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.
becomes
solving the output equation for and substituting in the state equation results in
The advantage of this is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of . 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.
For a strictly proper system D equals zero. Another fairly common situation is when all states are outputs, i.e. y = x, which yields C = I, the Identity matrix. This would then result in the simpler equations
This reduces the necessary eigendecomposition to just .
In addition to feedback, an input, , can be added such that .
becomes
solving the output equation for and substituting in the state equation results in
One fairly common simplification to this system is removing D, which reduces the equations to
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:
where
The state equation would then become
where
The controllability test is then
which has full rank for all and . This means, that if initial state of the system is known (, , ), and if the and are constants, then there is a force that could move the cart into any other position in the system.
The observability test is then
which also has full rank. Therefore, this system is both controllable and observable.
The more general form of a state-space model can be written as two functions.
The first is the state equation and the latter is the output equation. If the function is a linear combination of states and inputs then the equations can be written in matrix notation like above. The argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).
A classic nonlinear system is a simple unforced pendulum
where
The state equations are then
where
Instead, the state equation can be written in the general form
The equilibrium/stationary points of a system are when and so the equilibrium points of a pendulum are those that satisfy
for integers n.
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field whose value at a point gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of . If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function may be defined by:
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
Controllability is an important property of a control system and plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
In mathematics, a system of linear equations is a collection of one or more linear equations involving the same variables. For example,
In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and the determinant are often referred to simply as the Jacobian in literature.
In the mathematical field of differential geometry, a metric tensor is an additional structure on a manifold M that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p, and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p.
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable.
In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by applying a discontinuous control signal that forces the system to "slide" along a cross-section of the system's normal behavior. The state-feedback control law is not a continuous function of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward an adjacent region with a different control structure, and so the ultimate trajectory will not exist entirely within one control structure. Instead, it will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode and the geometrical locus consisting of the boundaries is called the sliding (hyper)surface. In the context of modern control theory, any variable structure system, like a system under SMC, may be viewed as a special case of a hybrid dynamical system as the system both flows through a continuous state space but also moves through different discrete control modes.
In multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals.
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically:
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.
Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in pre-determined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. The system must be considered controllable in order to implement this method.
Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form
A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the compartments of a system. Sometimes, the physical system that we try to model in equations is too complex, so it is much easier to discretize the problem and reduce the number of parameters. Each compartment is assumed to be a homogeneous entity within which the entities being modeled are equivalent. A multi-compartment model is classified as a lumped parameters model. Similar to more general mathematical models, multi-compartment models can treat variables as continuous, such as a differential equation, or as discrete, such as a Markov chain. Depending on the system being modeled, they can be treated as stochastic or deterministic.
In systems theory, a realization of a state space model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of (time-varying) matrices such that
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface. It is illustrated in the figure, where the direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule. For the right hand the fingers circulate along ∂Σ and the thumb is directed along n.
In systems and control theory, the double integrator is a canonical example of a second-order control system. It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input .
Variance-based sensitivity analysis is a form of global sensitivity analysis. Working within a probabilistic framework, it decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. For example, given a model with two inputs and one output, one might find that 70% of the output variance is caused by the variance in the first input, 20% by the variance in the second, and 10% due to interactions between the two. These percentages are directly interpreted as measures of sensitivity. Variance-based measures of sensitivity are attractive because they measure sensitivity across the whole input space, they can deal with nonlinear responses, and they can measure the effect of interactions in non-additive systems.