State space representation
The state space representation is one of several known types of the system description of a dynamic transmission system. The state space model is considered remarkable engineering skills appropriate method of analysis and synthesis of dynamic systems in the time domain and is particularly efficient in control engineering treatment of multivariable systems, nonlinear and time-varying transmission systems. In this case, all relations of the state variables, the input variables and output variables are represented in form of matrices and vectors. The state space model is represented by two equations - described - the state first order differential equation and the output equation.
- 3.1 Control Normal Form
- 4.1 Model of a transmission system
- 4.2 controllability
- 4.3 observability / observer
- 4.4 state feedback controller with observer
- 5.1 state regulator
- 5.2 state controller with superimposed control loop
- 6.1 Description of linear systems
- 6.2 Linear equations of state
- 6.3 Nonlinear Equations of State
- 6.4 similarity transformation
- 6.5 Transfer function
- 6.6 General solution in the time domain
- 6.7 Normal Forms 6.7.1 control normal form
- 6.7.2 Observation normal form
- 6.7.3 Canonical normal form
Basic System Description in the state space
The well-known since the 1960s, the theory of the state space comes from the USA by the mathematician and Stanford University teacher Rudolf E. Kalman. It is about the same time emerged with the onset of powerful digital computers, which are essential for dealing with the state space representation.
In higher education, the engineering disciplines of automation, mechatronics, electrical, etc. especially becoming increasingly similar in control engineering, the state-space representation of a larger range. Thus, the state space representation is considered after presentation of some academics have known in the past as a major technological boost to the aerospace industry as the Apollo program was completed in 1969 flight to the moon.
The term state space representation is defined as the description of a dynamic transmission system through its state variables ( = state variables ). The descriptive system of differential equations of order n with n concentrated energy storage is decomposed into n first -order differential equations and placed in a matrix / vector representation.
The state variables describe the physical energy content of the storage elements contained in a technical dynamic system. You mean, for example, voltage on a capacitor, current in an inductor, in a spring-mass system, the potential and kinetic energy components. The number of state variables of the state vector, the dimension of the state space. In the state vector at any time t (0) all of the dynamic information transmission system are included.
Essential concepts for the understanding of the description of a communication system in the state space are the state space model, and the normal form used, according to the state equations and associated matrices / vectors are designed. The state space model can be created for non- jump capable systems directly from the coefficients of the differential equation describing the system or the associated transfer function.
After the signal flow diagram of the control normal form can be formed with the help of the returned state variables dynamically advantageous state control loop, which can be simulated by numerical calculation without matrix representation of all the available signal parameters.
Overview of system descriptions
In classical control theory before the 1960s had the analysis and calculation of control equipment in the time domain only of less importance than the methods in the frequency and s- area, such as the Laplace transform, the frequency response and the root locus method. This linear time-invariant transfer elements were treated with constant coefficients mainly. Nonlinear systems have been linearized.
To understand the theory of the state space representation of the system descriptions following knowledge is required:
- Ordinary Differential Equations of a transmission system
- Description of linear systems in the complex frequency domain
- Numerical description of linear and nonlinear systems
Definition of the state of a transmission system
While the above description characterizes the system response of a system, is a description of a system in the state space, the current state of the system at a certain time t = 0
Physically, the state of a dynamic system is determined by the energy content of the energy present in the system memory. The state variables describe the energy content of the storage elements contained within the system. You can not change abruptly at excitation of the system.
The value of the state variables at that particular time t is the status of the system and is summarized by the vector.
The behavior of the transmission system is at any time at the time t = 0 completely given for t> 0 if
- The mathematical model of the transmission system is known,
- The initial values of the energy storage are known and
- The input variables of the system are known.
It follows:
With knowledge of the system state and all forces acting on the system signal variables, the future system behavior for t can be > 0 determined in advance.
The number of the state variable of the dimension of the state space.
Definition: the state space, vector space, phase space, phase portrait
In the German -speaking world, the concept of state space representation has been created for the older and even today valid concept system described in state space only after the 1970s.
According to the literature of control theory the single term " state space " of a dynamic transmission system is defined as follows:
Because of the similarity of the definition of " state space " with the mathematical concepts vector space and phase space we can assume that the concept of state space is the result of the primary term " state of a system " and the term " phase space ".
The system behavior of a dynamic transmission system in the state space using the example of a delay higher-order system can be particularly graphically vividly represented by:
- Graphical representation of the phase portraits
- Recording the progression of the state variable F ( t)
Example state variables and equations of state for a PT2 A vibrating element
Standard transfer function of a vibration member ( PT2 element ) with complex conjugate poles ( PT2KK element):
The associated differential equation is determined by conversion using the inverse Laplace transform:
In the literature, the coefficients of the derivatives of y (t) ( here T ², DT 2 ) are shown to unify with the letter a, for the right side of the derivatives of u (t ) with b and numbered consecutively:
The highest derivative is exempted from the coefficients in which all terms of the equation are divided by and is resolved by:
The block diagram shown in the structure diagram corresponds to the classical variant of the solution of a differential equation with the help of analog computing technology. This method has long been known. Of course, the interest was only the behavior of the output variable y (t).
Each derivative of the output variable y ( t) is subjected to integration. Each state variable is fed back with the respective coefficient to the input and subtracted from the input value u (t).
A differential equation of nth order needed to solve s integrations. After the block diagram for the solution of the differential equation of 2nd order is two state variables arise as outputs of the integrators. By substitution of the derivatives of y (t) by the designation of the state variables x ( t) used as follows:
This is the differential equation with the introduced new names of the state variables:
The transformation of the system of differential equations describing the nth-order coupled differential equations in n- 1 order is as follows:
If you imagine, according to the block diagram, for example, the state variable from the output of the integrator at the input of the integrator same offset front, then the derivation is from.
From this follow the state differential equations first order:
The state variables and form the so-called state vector.
These equations are written as a vector differential equations in matrix form as follows:
And the output equation:
There are several signal flow diagrams that lead to the solution of the differential equation and the determination of the state variables. The quotient can be according to the block diagram of the vibration member left the subtraction, it can be right away, or the equation can be transformed so that the highest derivative has the coefficient 1. All these measures lead to the same result for the output variable y ( t). However, this does not apply to the definition of the coefficients of the state variables.
For transmission systems with poles and zeros, there is therefore a uniform normal form, preferably the " control normal form " to represent the signal flows.
State space model
Wherein the state space representation is assumed to be the state space model.
The block diagram of the signal flow diagram of the state space model shows a SISO transmission system with an input signal u (t ) and an output signal y ( t) in a general representation for a linear transmission system with n differential equations first order. It corresponds to the system representation of the control normal form. Instead of a differential equation system of order n occurs a derivative of the state vector first order. The input of the integrator is derived from the n-dimensional state vector and the output of the state vector.
The so-called equations of state for the derivative of the vector and the output variable y (t ) of a Eingrößensystems:
Can be read directly from the block diagram of the state space model.
The block diagram of the state space model has a uniform shape, but presented in different ways as a single or multi-variable system. In the multivariable system take the place of the scalar input and output variables u (t ) and y ( t) are the vectors and. The signal flows of matrices and vectors are shown in the block diagram by double lines.
Summary of state space model:
- State space model and block diagram
- System matrix
- Linear and non- linear systems
- State differential equation and output equation
Indexing:
- Matrices = uppercase with underscore
- Vectors = lowercase with underscore
- Transpose of vector representation, example:
Linear transmission systems with multiple input and output variables can be described with linear state differential equations.
The output equations for linear systems have the following forms:
These equations represented in vector notation you can also play in matrix notation.
State differential equations of multivariable systems
Output equations of multivariable systems:
SISO systems have only one input variable u ( t) and output y (t). The input matrices and output matrices will become the input vector and output vector.
State differential equations of the single variable
Output equation of the single variable:
Normal forms in the state space
In the state descriptions with normal forms the state equations take on a particularly simple and practical forms for specific calculations. For the normal forms is assumed that the system or the transmission system by the linear differential equation or associated transfer function.
Among the best known normal forms include:
- Control normal form (also with Frobenius form, control normal form or standard form referred to 1 ),
- Observation normal form
- Canonical normal form
The normal forms are recognizable in the system matrix by the local position of the coefficients.
Control normal form
The signal structure of the control normal form presents itself as an analog continuous-time system is that the input u ( t) is the solution of the differential equation y ( t) reproduces and simultaneously shows the state variables.
The block diagram of the control normal form shows the implementation and solution of the differential equation in the physical analog signal flows of the state variables, including the output for a given input size. They can be seen as an evolution of the well-known in the analog computing art methods for the solution of a differential equation consider the nth order with n integrators. The signal flows can be determined directly by numerical calculation for arbitrary input signals and plotted with knowledge of the coefficients of the state variables.
The simplified example of the block diagram of signal flow plan of 2nd order is a transmission system that includes only poles. For any system with poles and zeros in the control normal form of the signal flow diagram for the derivatives of the input variable u ( t) must be extended so that the terms to the output variable y ( t) add up.
Transfer function and associated differential equation in polynomial
The transfer function of a linear transmission system in polynomial is defined as the ratio of output signal to input signal as a function of the complex frequency, s is formed under the condition that the initial conditions of the energy accumulators of the output Y (s ) are set to zero:
Where n = number of poles and m = number of zeros of the system mean:
- N> m: This is the normal case in control theory, that is, the number of poles n is greater than that of the zeros of m. The system is not able to jump.
- M = n: This relationship with the same number of poles and zeros occurs only in exceptional cases. The system is able to jump, that is a step change in the input variable will be transmitted without delay to the output.
- M> n: These systems can not be treated with the state space representation. They are also not technically feasible.
The corresponding differential equation of the transfer function is given by the inverse Laplace transform.
The highest degree of discharge of indicates the number of memory elements of the route again.
State variables arise from the poles of the transmission system
The state variables of a linear system of order n with n energy storage always arise from the poles. Has the transmission system also zeros - ie differentiating shares - so the state variables are added to the coefficients of the derivatives of the input variable u (t ) to the output y (t). Statement from systems theory: The poles of a transfer function determine the speed of the system of movement and stability. The zeros of a transfer function only affect the amplitudes of the system.
Indexing the derivatives of y (t)
Because the output of the transmission system will already be designated y ( t), the differential equation describing the system with the derivatives of y (t) to be indexed. The differential equation is replaced instead of the symbol y (t ) the icon and u ( t) is inserted through. In the state space representation disappear derivatives of y (t) and replaced by the state variable x (t).
Thus, the state variables
In the block diagram of the control normal form of the derivatives are replaced by the state variables, so that no longer appear.
Definition pole-zero ratio and coefficients of the differential equation
The control normal form is valid for linear systems with n poles and m zeros to n → m.
With the help of numerical discrete-time calculation methods to solve the differential equation y ( t) and the evolution of the state variables for a given input signal u ( t) can be easily determined.
The corresponding matrix representation for a common in control engineering leaps and incompetent system of order n is in control normal form with the following conditions:
The descriptive system transfer function or the associated differential equation can be transformed so that the coefficient of the highest derivative of y (t) equal to 1. All coefficients are arranged by dividing and new.
Example: state variables for a transmission system of 4th order
The transfer function of a transmission system, for example, the 4th order ( with the derivatives of the input 3rd order ) is the allowable for control technology restriction n> m and the coefficient of the highest derivative of y (t ) ::
The associated differential equation is for a transmission system of 4th order then the permissible limits m
From the differential equation ( substituting the derivatives of y (t) by x (t ) ) result by the known scheme following state equations:
These equations can be converted for the state space representation in matrix notation as state differential equations for SISO systems always the same scheme:
Output equations for SISO systems:
Under the state-space model in the control normal form is defined as a uniform form of matrix representation with the following advantageous properties:
For the analysis, synthesis and control of real transmission systems ( usually Streckeen ), which are generally present as a hardware system, a mathematical model of the system is required.
Models in the form of differential equations describe the temporal behavior of the system accurately.
Are these differential equations or transfer functions associated not given, the temporal behavior of a hardware system can ( system identification Experimental ) are determined by using the test signals by experimental identification measures.
In the basic procedure of the identification algorithm for the model parameters is varied until, for a given input signal u ( t) is the difference of the output variables y (t) - yModell (t) within an arbitrary time sequence of measured original output to said model output approximately disappears.
The model defines the structure of a signal flow plan, from which the state variables can be derived.
If the system has n energy stores, so the n state variables can be summarized by the state variable vector.
Are the input value u (t) and the coefficients of the state variables of the track is known, the output y (t) can be calculated.
The transmission system (controlled system ) must be controllable.
All state variables must be available.
Pole-zero compensation in the state space is not allowed, because a loss of information occurs.
While Eingrößensystemen the problem of controllability and observability rarely arises because these systems are easily seen through, a test according to known rules may be necessary with coupled multivariable systems.
In the state of control, all the state variables are returned to the input of the system.
For the implementation of a state control all state variables must be available.
This condition is fulfilled when the manipulated variable of the controller acts on all state variables.
A system can be controlled, if it can be brought from an arbitrary initial state after a finite time to a desired final state.
In general, for the controllability of the signal relative sizes:
All states ( state variables ) of a controllable system, including the system can be controlled.
State regulations require all of the state variables of the transmission system.
The state variables are determined by measurement of the controlled system.
Is this the case given, this corresponds to the " full observability ".
Often can be measured for technical or commercial reasons, not all state variables.
Therefore, individual non-measurable state variables from the known and existing input and output variables of the controlled system are calculated.
State observer that perform this task, additional control systems.
Reconstruct state variable from the variation of the input and output variables in a model of the controlled system.
State observer can be realized only when the observed system is observable, which is the case for the vast majority of technical control systems.
A linear transmission system is observable when, by measuring the output variables y ( t) of the initial state of the state vector can be determined after a finite time.
The input variable u (t) must be known.
For the realization of the state observer, the separation principle is applied.
It allows the separate design of the state feedback and observation.
Such a procedure can be realized with the " Luenberger observer ."
The control system and the observer with the model control system are connected in parallel at the input u (t).
The output of both systems is monitored and used to correct the model control system.
The observer requires the most accurate model of the plant.
The structure of the controlled system and identification techniques using the jump or impulse response of the controlled system, a model of the controlled system can be formed that is associated in most cases even with small errors.
According to the method with the Luenberger observer of controlled- output y ( t) is compared with the model output and returned additive via a control loop to the input of the model, so that the output of the controlled system and the output of the model within a settling time are identical.
It is assumed that the unknown state vector and the determined model state vector are then almost identical.
For state control loop, the determined state vector with weighting factors for the desired dynamic behavior of the state of the control circuit of the reference variable w (t ) is subtracted.
In single-loop standard control loops usually the output variable y (t ) of the control circuit of the reference variable w (t ) is subtracted and then as the error e (t ) = w (t ) - y ( t) is fed to the controller.
This procedure is referred to in connection with the treatment of systems in the state space to output feedback.
A conventional controller design strategy for controlled systems with output feedback is relatively simple:
By pole-zero compensation of controller and control loop and open-loop transfer function of the overall control loop is simplified, that is, the order of the differential equation of the controlled system is reduced, in which the values of the zeros of the controller are set to the values of the pole of the track.
A I component of the controller avoids a permanent deviation, but adds an additional pole.
For optimizing parameters, the loop gain K. The step response of the controlled variable runs as a function of K asymptotically to strong overshooting until the setpoint is reached.
The state variables of a mathematical model of a controlled system can be determined from an ordinary system of differential equations describing.
Basis of the solution of the differential equation is the signal flow diagram of the graphical representation of the control normal form.
The terms of the derivative of the output y (t) are respectively integrated and fed back with the respective coefficient to the input system.
For each derivative y ( t) is the name of the state variables x (t ) is introduced as follows:
The time course of the state variables as a result of an input jump u ( t) = 1 in the model shows the advantage of treating the system in the state space with respect to a classical " output feedback " of the system.
The state variables x (t ) appear earlier in time than the output variable y ( t).
This behavior is used in the loop condition by the state variables are attributed to a target-actual difference with the reference variable w (t).
Simulations of a state control loop can be performed easily with a good model of the controlled system on a programmable calculator.
The description of the signal flow plan of the controlled system and the controller in state space can be done both in the form of matrices as well as difference equations.
Depending on the height of the order of the differential equations, all state variables are supplied to a state regulator, which acts on the input of the state space model of the controlled system.
With the return of all the state variables results in a more circuitous creeping control loop.
The linear state controller evaluates the individual state variables of the controlled system with factors and summing the products thus formed state to a desired-actual value comparison.
It is in this state, the controller is not a P-controller, although such an impression could be noisy signal flow diagram.
By recirculated to the controller state variables with weighting factors again and run the calculation circuit for the solution of the differential equation with n integrators and form new circuit variables, thus forming derivative action.
Therefore, corresponding to the effect of the returned state variables depending on amount of order n of the differential equation of the path of a controller.
As a design strategy for the determination of the weighting factors of the state controller applies the pin assignment ( pole placement ) of the closed loop
Also empirical settings of a model loop are easily possible.
By the series connection of the integrators only the state variable x1 (t) = y ( t) is a stationary variable if the input u (t ) is constant.
All other state variables - provided a stable controlled system - tend to zero.
After adjustment and optimization of the factor k1, a stable control loop of certain damping results in a proportional error of the controlled variable y (t ) with respect to w ( t).
The other factors of the state variables are sequentially set in order to optimize the transient response, for example.
A pre-filter before the target-actual comparison corrects the static error between w (t ) and y (t).
Prerequisite for the return of the state variables:
With the state controller results in the following control properties of a control loop:
The state controller in the illustrated property useful for the understanding of its benefits.
It allows in his behavior as a PD controller, a higher loop gain than in a control loop with an output feedback.
In this case, no differentiating component is included within the state of the control loop.
In a comparison with a standard PD2 controller and output feedback, and otherwise the same circuit damping, the same control path and disturbance arising for this controller significant disadvantages like giant manipulated variables, very bad noise suppression and manipulated variable limits of > 2 (= 200%) totzeitähnliches minimum phase behavior.
Depending on the requirements with respect to control error and interference at the output of the controlled system can be inferior in comparison with a conventional PID controller standard.
Remedy these disadvantages, creates a layered with a PI regulator control circuit.
Thus, the state controller has significantly better properties.
The use of state regulators ultimately depends on a cost -benefit assessment.
For control tasks with systems in the state space, the introduction of a superimposed control loop for the state control circuit may be required with output feedback.
This means that all stationary problems for the conformity of the reference variable must be switched off by the control variable and constant interference components.
It is recommended to use a PI controller.
According to the transfer function of the PI controller to the product presentation is the controller of the components I-element and the PD element.
This control can only be a temporary deviation to, provided that the reference variable and a possible disturbance are constant.
The Pd component is a deceleration component ( PT1 ) of the state control circuit can be compensated.
Signal noise of the PD element is reduced by the I element.
The following parameters are observed for the interpretation of superimposed PI - state control loop:
A state feedback controller with superimposed PI control loop has over a conventional well -optimized control loop, each with the same control path and the same transient dynamic unique advantages.
Advantages:
Disadvantages in general:
Continuous-time linear systems through linear differential equation of order n
Described.
If the coefficients and are all constant, the Laplace transform is executed, and it is the transfer function
A differential equation of order n can be used in a system of n differential equations first order
Be transferred.
Discrete-time linear systems by linear difference equation of order n
Described.
If the coefficients and are all constant, the z-transform is executed, and it is the transfer function
A difference equation of order n can be used in a system of n differential equations first order
Be transferred.
For continuous-time systems, the basic equations are linear in vector form:
About the matrices and the concatenation of the individual states, including the number of hits on the control variables ( input variables ) are represented.
The matrix is referred to as a matrix system, as the control matrix.
The observation matrix describes the impact of the system on the output.
The passage matrix describes the through portions of the system, it is not in leaps and capable systems zero.
An important special case systems with an input and an output variable represents ( SISO single input, single-output systems).
Here are and vectors and a scalar.
There are then often the symbols, and used.
In many cases, instead of a continuous course interested only the system state at discrete points, for example at the sampling control by a digital computer.
In this case, instead of a vector-valued function of time, a sequence of vectors.
Then replaced by a difference equation in place of the state differential equation.
The types of linear basic equations:
For the last two cases it was assumed that the initial state of the system (see Differentiation of the Laplace transform or Z-transform of the set of differences ).
The discrete-time state space description is from the continuous form by means of discretization on a fixed time step t in the form
Won.
Gilt yields the integral
.
The discrete form is particularly suitable for calculations in real time.
In real time, the output equation is calculated first, and only then the state difference equation to determine the conditions for the next calculation step.
The continuous time representation is, however, good for simulations without real-time claims by numerical integration.
The accuracy can be influenced here by the choice of the integration process and the adaptation of static or dynamic step size.
Of central importance is the system matrix, from which the eigenvalues , and thus the system dynamics and their stability can be derived is ( characteristic polynomial ).
If the penetration matrix no zero matrix, the system inputs have simultaneous influence on the outputs, which can lead to an algebraic loop.
A, B, C, D is constant, the system is linear and time-invariant, i.e. a so-called LTI system.
A nonlinear system of order n may be used as a system of non-linear differential equations, the first -order
Or in more compact vector notation
Be written.
For the rest point
If the deviation of the system from the point of rest, then applies
And
The linearized representation
With the Jacobian matrices, and results from a multi-dimensional linearized Taylor expansion around the point of rest.
The state space representation is not unique.
On the same system, there are an infinite number of state-space representations.
Instead of the usual state variables you can also use a new set of state variables, if you can describe through.
Wherein a regular, linear transform matrix, i.e., must be described by adding inputs or without derivatives.
We then have:
The new state space representation describes the same system.
It is therefore to be understood that all system characteristics in the transformation remains unchanged.
The " transfer function " of a continuous time-invariant state-space model can with vanishing initial conditions (x ( 0) = 0) can be derived in the following way:
By the Laplace transform is obtained
Which is substituted in the output equation
And the transfer function yields
This corresponds to the identity matrix.
The general solution in the time domain is obtained by:
Difficulties can prepare the matrix exponential function, which is defined analogously to the scalar exponential function by the power series
To be able to specify here a closed expression, it is helpful to transform by principal axis transformation to diagonal form.
For a diagonal matrix of the form
Then yields the matrix exponential to
Normal forms are used to clearly highlight structural properties of a system.
Often has a system in the state space representation of state variables, which do not manifest themselves in the transmission behavior of the system.
It may be, for example, that cut poles and zeros, so that they have no influence on the transfer function.
This case is called a non- minimum implementation of the system, and this leads to that the system is either not controlled, can not be observed, or neither controllable nor observable.
The given transfer function can be converted using the following approach in a state space representation.
The given transfer function is multiplied out in the numerator and denominator factors
At this transfer function in the frequency domain in the time domain include the differential equation ( ODE ):
: The following equations of state representation by control normal form - From this ODE are obtained for the ZR
The coefficients can now be easily added directly to the state matrices:
The transformation matrix follows the controllability
.
If the system is controllable.
Then from
The transformation matrix
Are formed.
The dissolved differential equation for
And 4 results integrated times
From this, the state variables
And the output equation
Derived.
Inserting results
Or in matrix form
The transformation matrix follows from the observability matrix
When the system is observable.
Then from
The transformation matrix
Are formed.
Has the transfer function of simple real poles can be a partial fraction expansion of the form
Be performed.
from
Result from reverse transformation the equations of state
And the output
.
In matrix notation
And
The state equations are decoupled in this case.
The transformation matrix is calculated from the associated with the eigenvalues of the system matrix of eigenvectors in the form of
Written.
Controllability and observability of transmission systems
Model of a transmission system
Controllability
Observability / observer
State feedback controller with observer
Regulation in the state space
State regulator
State regulators with superimposed control loop
Mathematical concentrate of rules and equations in the state space
Description of linear systems
Linear equations of state
Nonlinear equations of state
Similarity transformation
Transfer function
General solution in the time domain
Normal Forms
Control normal form
Observation normal form
Canonical normal form