State observer
Encyclopedia
In control theory
, a state observer is a system that models a real system in order to provide an estimate of its internal state
, given measurements of the input
and output
of the real system. It is typically a computer-implemented mathematical model.
Knowing the system state is necessary to solve many control theory
problems; for example, stabilizing a system using state feedback
. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable
, it is possible to fully reconstruct the system state from its output measurements using the state observer.
where, at time
, 
is the plant's state; 
is its inputs; and 
is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current state and the current inputs. (Although these equations are expressed in terms of discrete
time steps, very similar equations hold for continuous
systems). If this system is observable
then the output of the plant,
, can be used to steer the state of the state observer.
The observer model of the physical system is then typically derived from the above equations. Additional terms may be included in order to ensure that, on receiving successive measured values of the plant's inputs and outputs, the model's state converges to that of the plant. In particular, the output of the observer may be subtracted from the output of the plant and then multiplied by a matrix
; this is then added to the equations for the state of the observer to produce a so-called Luenberger
observer, defined by the equations below. Note that the variables of a state observer are commonly denoted by a "hat":
and 
to distinguish them from the variables of the equations satisfied by the physical system.
The observer is called asymptotically stable if the observer error
converges to zero when 
. For a Luenberger
observer, the observer error satisfies
. The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix 
has all the eigenvalues inside the unit circle.
For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix
.
The observer equations then become:
or, more simply,
Due to the separation principle
we know that we can choose
and 
independently without harm to the overall stability of the systems. As a rule of thumb, the poles of the observer 
are usually chosen to converge 10 times faster than the poles of the system 
.
are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when 
is a Hurwitz matrix
).
For a continuous-time linear system
,
where
, the observer looks similar to discrete-time case described above:
.
The observer error
satisfies the equation
.
The eigenvalues of the matrix
can be made arbitrarily by appropriate choice of the observer gain 
when the pair 
is observable, i.e. observability
condition holds. In particular, it can be made Hurwitz, so the observer error
when 
.
is high, the linear Luenberger observer converges to the system states very quickly. However, high observer gain leads to a peaking phenomenon in which initial estimator error can be prohibitively large (i.e., impractical or unsafe to use). As a consequence, nonlinear high gain observer methods are available that converge quickly without the peaking phenomenon. For example, sliding mode control
can be used to design an observer that brings one estimated state's error to zero in finite time even in the presence of measurement error; the other states have error that behaves similarly to the error in a Luenberger observer after peaking has subsided. Sliding mode observers also have attractive noise resilience properties that are similar to a Kalman filter
.
where
. Also assume that there is a measurable output 
given by
There are several non-approximate approaches for designing an observer. The two observers given below also apply to the case when the system has an input. That is,
.
, like the one used in feedback linearization
)
such that in new variables the system equations read
The Luenberger observer is then designed as
.
The observer error for the transformed variable
satisfies the same equation as in classical linear case.
.
As shown by Gauthier, Hammouri, and Othman
and Hammouri and Kinnaert, if there exists transformation
such that the system can be transformed into the form
then the observer is designed as
,
where
is a time-varying observer gain.
where there is no difference between the estimated output and the measured output. The non-linear gain used in the observer is typically implemented with a scaled switching function, like the signum
(i.e., sgn) of the estimated–measured output error. Hence, due to this high-gain feedback, the vector field of the observer has a crease in it so that observer trajectories slide along a curve where the estimated output matches the measured output exactly. So, if the system is observable
from its output, the observer states will all be driven to the actual system states. Additionally, by using the sign of the error to drive the sliding mode observer, the observer trajectories become insensitive to many forms of noise. Hence, some sliding mode observers have attractive properties similar to the Kalman filter
but with simpler implementation.
As suggested by Drakunov, a sliding mode observer can also be designed for a class of non-linear systems. Such an observer can be written in terms of original variable estimate
and has the form
where:
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, a state observer is a system that models a real system in order to provide an estimate of its internal state
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
, given measurements of the input
Input/output
In computing, input/output, or I/O, refers to the communication between an information processing system , and the outside world, possibly a human, or another information processing system. Inputs are the signals or data received by the system, and outputs are the signals or data sent from it...
and output
Output
Output is the term denoting either an exit or changes which exit a system and which activate/modify a process. It is an abstract concept, used in the modeling, system design and system exploitation.-In control theory:...
of the real system. It is typically a computer-implemented mathematical model.
Knowing the system state is necessary to solve many control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
problems; for example, stabilizing a system using state feedback
Full state feedback
Full state feedback , 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,...
. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable
Observability
Observability, in control theory, 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. The concept of observability was introduced by American-Hungarian scientist Rudolf E...
, it is possible to fully reconstruct the system state from its output measurements using the state observer.
Typical observer model
The state of a physical discrete-time system is assumed to satisfywhere, at time
, is the plant's state; is its inputs; and is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current state and the current inputs. (Although these equations are expressed in terms of discreteDiscrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
time steps, very similar equations hold for continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
systems). If this system is observable
Observability
Observability, in control theory, 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. The concept of observability was introduced by American-Hungarian scientist Rudolf E...
then the output of the plant,
, can be used to steer the state of the state observer.The observer model of the physical system is then typically derived from the above equations. Additional terms may be included in order to ensure that, on receiving successive measured values of the plant's inputs and outputs, the model's state converges to that of the plant. In particular, the output of the observer may be subtracted from the output of the plant and then multiplied by a matrix
; this is then added to the equations for the state of the observer to produce a so-called LuenbergerDavid Luenberger
David G. Luenberger is a mathematical scientist known for his research and his textbooks, which center on mathematical optimization. He is a professor in the department of Management Science and Engineering at Stanford University.-Biography:...
observer, defined by the equations below. Note that the variables of a state observer are commonly denoted by a "hat":
and to distinguish them from the variables of the equations satisfied by the physical system.The observer is called asymptotically stable if the observer error
converges to zero when . For a LuenbergerDavid Luenberger
David G. Luenberger is a mathematical scientist known for his research and his textbooks, which center on mathematical optimization. He is a professor in the department of Management Science and Engineering at Stanford University.-Biography:...
observer, the observer error satisfies
. The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix has all the eigenvalues inside the unit circle.For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix
.The observer equations then become:
or, more simply,
Due to the separation principle
Separation principle
In control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal observer for the state of...
we know that we can choose
and independently without harm to the overall stability of the systems. As a rule of thumb, the poles of the observer are usually chosen to converge 10 times faster than the poles of the system .Continuous-time case
The previous example was for an observer implemented in a discrete-time LTI system. However, the process is similar for the continuous-time case; the observer gains are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when is a Hurwitz matrixHurwitz matrix
-Hurwitz matrix and the Hurwitz stability criterion:In mathematics, Hurwitz matrix is a structured real square matrix constructed with coefficientsof a real polynomial...
).
For a continuous-time linear system
,where
, the observer looks similar to discrete-time case described above:.The observer error
satisfies the equation.The eigenvalues of the matrix
can be made arbitrarily by appropriate choice of the observer gain when the pair is observable, i.e. observabilityObservability
Observability, in control theory, 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. The concept of observability was introduced by American-Hungarian scientist Rudolf E...
condition holds. In particular, it can be made Hurwitz, so the observer error
when .Peaking and other observer methods
When the observer gain is high, the linear Luenberger observer converges to the system states very quickly. However, high observer gain leads to a peaking phenomenon in which initial estimator error can be prohibitively large (i.e., impractical or unsafe to use). As a consequence, nonlinear high gain observer methods are available that converge quickly without the peaking phenomenon. For example, sliding mode controlSliding mode control
In control theory, sliding mode control, or SMC, is a nonlinear control method that alters the dynamics of a nonlinear system by application of 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...
can be used to design an observer that brings one estimated state's error to zero in finite time even in the presence of measurement error; the other states have error that behaves similarly to the error in a Luenberger observer after peaking has subsided. Sliding mode observers also have attractive noise resilience properties that are similar to a Kalman filter
Kalman filter
In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...
.
State observers for nonlinear systems
Sliding mode observers can be designed for the non-linear systems as well. For simplicity, first consider the no-input non-linear system:where
. Also assume that there is a measurable output given byThere are several non-approximate approaches for designing an observer. The two observers given below also apply to the case when the system has an input. That is,
.Linearizable error dynamics
One suggested by Kerner and Isidori and Krener and Respondek can be applied in a situation when there exists a linearizing transformation (i.e., a diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
, like the one used in feedback linearization
Feedback linearization
Feedback linearization is a common approach used in controlling nonlinear systems. The approach involves coming up with a transformation of the nonlinear system into an equivalent linear system through a change of variables and a suitable control input...
)
such that in new variables the system equations readThe Luenberger observer is then designed as
.The observer error for the transformed variable
satisfies the same equation as in classical linear case..As shown by Gauthier, Hammouri, and Othman
and Hammouri and Kinnaert, if there exists transformation
such that the system can be transformed into the formthen the observer is designed as
,where
is a time-varying observer gain.Sliding mode observer
As discussed for the linear case above, the peaking phenomenon present in Luenberger observers justifies the use of a sliding mode observer. The sliding mode observer uses non-linear high-gain feedback to drive estimated states to a hypersurfaceHypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
where there is no difference between the estimated output and the measured output. The non-linear gain used in the observer is typically implemented with a scaled switching function, like the signum
Signum
Signum is Latin for "sign" and may refer to:* Signum function or sign function in mathematics* Signum, a part of the female Lepidoptera genitalia* Signum , a 1995 by German industrial music artist P·A·L* Signum Framework...
(i.e., sgn) of the estimated–measured output error. Hence, due to this high-gain feedback, the vector field of the observer has a crease in it so that observer trajectories slide along a curve where the estimated output matches the measured output exactly. So, if the system is observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
from its output, the observer states will all be driven to the actual system states. Additionally, by using the sign of the error to drive the sliding mode observer, the observer trajectories become insensitive to many forms of noise. Hence, some sliding mode observers have attractive properties similar to the Kalman filter
Kalman filter
In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...
but with simpler implementation.
As suggested by Drakunov, a sliding mode observer can also be designed for a class of non-linear systems. Such an observer can be written in terms of original variable estimate
and has the formwhere:
- The
vector extends the scalar signum functionSign functionIn mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
to
dimensions. That is,
-
- for the vector
.
- The vector
has components that are the output function 
and its repeated Lie derivatives. In particular,
- where
is the ith Lie derivativeLie derivativeIn mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
of output function
along the vector field 
(i.e., along 
trajectories of the non-linear system). In the special case where the system has no input or has a relative degree of n, 
is a collection of the output 
and its 
derivatives. Because the inverse of the Jacobian linearizationLinearizationIn mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
of
must exist for this observer to be well defined, the transformation 
is guaranteed to be a local diffeomorphismDiffeomorphismIn mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
.- The diagonal matrixDiagonal matrixIn linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
of gains is such that
- where, for each
, element 
and suitably large to ensure reachability of the sliding mode.
- The observer vector
is such that
- where
here is the normal signumSignumSignum is Latin for "sign" and may refer to:* Signum function or sign function in mathematics* Signum, a part of the female Lepidoptera genitalia* Signum , a 1995 by German industrial music artist P·A·L* Signum Framework...
function defined for scalars, and
denotes an "equivalent value operator" of a discontinuous function in sliding mode.
The idea can be briefly explained as follows. According to the theory of sliding modes, in order to describe the system behavior, once sliding mode starts, the function
should be replaced by equivalent values (see equivalent control in the theory of sliding modes). In practice, it switches (chatters) with high frequency with slow component being equal to the equivalent value. Applying appropriate lowpass filter to get rid of the high frequency component on can obtain the value of the equivalent control, which contains more information about the state of the estimated system. The observer described above uses this method several times to obtain the state of the nonlinear system ideally in finite time.
The modified observation error can be written in the transformed states
. In particular,
and so
So:- As long as
, the first row of the error dynamics, 
, will meet sufficient conditions to enter the 
sliding mode in finite time. - Along the
surface, the corresponding 
equivalent control will be equal to 
, and so 
. Hence, so long as 
, the second row of the error dynamics, 
, will enter the 
sliding mode in finite time. - Along the
surface, the corresponding 
equivalent control will be equal to 
. Hence, so long as 
, the 
th row of the error dynamics, 
, will enter the 
sliding mode in finite time.
So, for sufficiently large
gains, all observer estimated states reach the actual states in finite time. In fact, increasing 
allows for convergence in any desired finite time so long as each 
function can be bounded with certainty. Hence, the requirement that the map 
is a diffeomorphismDiffeomorphismIn mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
(i.e., that its Jacobian linearizationLinearizationIn mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
is invertible) asserts that convergence of the estimated output implies convergence of the estimated state. That is, the requirement is an observability condition.
In the case of the sliding mode observer for the system with the input, additional conditions are needed for the observation error to be independent of the input. For example, that
does not depend on time. The observer is then
- where
- The observer vector
- where, for each
- The diagonal matrix
- where
- The vector
- for the vector