Boris.T.Fedosov
Rudny Industrial Institute,
Rudny, Kazakhstan
About author
UDC 681.51.01
F338
Generalisation of state equations on nonlinear multi-dimensional dynamic objects and the control systems having extention in space and units of transport delay is conducted. Generalisation is executed by incorporation of delay links, along with integrators, in a structure of the elementary dynamic, i.e. such which output variables are treated as self-maintained state variables.
The traditional mathematical description of the dynamic object in state variables includes the vector of a state equations linking velocities of a modification of state variables with actions on the object and values of state variables, and also a vector equation linking values of output variables of the object (or results of their measurings) with its state variables and actions on it [1 - 9]:
(1.1)
The system (1.1) is a system of the vector differential-algebraic equations of state variables of the multi-dimensional nonstationary nonlinear inertia-dynamic object of control concentrated in space (pointwise) .
From the equations (1.1) it is easy to see, that the description of the dynamic object without delays structurally contains only three types of operators: linear differentiating (purely dynamic, the inertia) and two inertialess nonlinear: a linking element and a composition unit:
(1.2)
The linear differentiating operator describes inertia because sets instantaneous velocity of a modification of a state variable, and, therefore, estimates a value a variable known by the current moment on some, let a small time slice forwards. It also should be treated as inertia, i.e. some predeterminancy of behaviour.
Fig. 1.1. The description of the inertia object and its structural model. The differential equation, represents relationship of cause and effect an action х and response y the elementary inertia link: action х leads to a modification of output variable y such, that a velocity this modification directly proportional to action . The integrator - model of the elementary, fundamental dynamic (inertia) unit. The structural model represents how the reason, action, will be converted to a consequence, output variable: the model of the elementary (fundamental) inertance ensures of action accumulation and saving
In a linearized object model the principle of superposition is valid and consequently the operator of a composition of variables represents their weighed total, and the communication statement becomes linear:
(1.3)
The equations of the dynamic object in state variables it is possible to present in an integrated kind, more evident for structural simulation:
(1.4)
The state equation describes a characteristic, interior inertance of the dynamic object. The output equation take into account disturbations to measuring a component of a vector of output variables.
The status and behaviour trend at least on an infinitesimal interval forwards of pure the inertia dynamic object is determined by a set of values of all state variables of the object in some instant and represented by an appropriate position of a figuring point in a multi-dimensional state space. As this information for the inertia object without delay complete, co-ordinates of any trajectory point (a figuring point) can be considered as initial conditions for integration of state equations, i.e. for determination of all subsequent motion trajectory of a figuring point, an estimate of behaviour of the dynamic object under external actions or for lack of those.
By way of illustration it we will result phase portraits (motions trajectories of figuring points of objects in a two-dimensional state space) for model of the free oscillating system with differing initial conditions:
Fig. 1.1. Phase portraits of the free inertia oscillating system at the different initial conditions matching to the same phase trajectory coincide, i.e. co-ordinates of any point of a phase trajectory can be considered as the initial conditions completely defining further free behaviour of the object
Thus, the behaviour pointwise (it is exclusive the inertia, not having delay components) dynamic objects is completely described by state and output equations, and also the initial conditions representing values of all state variables of the object in some moment of time, and represented by some trajectory, and the current object state is characterised by a point in multi-dimensional space of state variables.
The count of links of delay in models of objects as second, self-maintained kind of the elementary dynamic units, along with inertia (integrators), allows to describe dynamic objects in state variables uniformly practically any complexity and on this basis to carry out their analysis and optimisation.
Presence of delay components in some branches of model of the dynamic object essentially, and often in principle, changes dynamic properties of the object in comparison with the object without delay components. Therefore the state space matching only to output variables of the inertia units (integrators) is not to the full sets a status and behaviour of the object having links of delay.
The delay component of the dynamic object, also, as well as the inertia, it is necessary to consider as dynamic, and its output variable - as a separate state variable.
The basis for reference of a link delaying a signal on time finite interval to the elementary dynamic supports against a likeness and distinctions of two kinds of the elementary dynamic units of models of real objects and consists in the following.
Exterior difference consists that the inertia unit is described by the elementary differential equation while retarding - by algebraic.
The term "dynamic" attribute to the objects which behaviour under an external action can be predicted at least on an infinitesimal interval. The inertia unit, the integrator which is traditionally considered as the only thing dynamic, to such demand responds. But the same demand is responded also by a delay link if the previous history of action on him is known. In that case the delay link allows to determine rigidly behaviour of its output variable on time finite interval forwards. Thus, the delay link can be attributed to the dynamic.
On the other hand, the delay link matches in real objects or to transposition of substances ("transportation delay"), or delay of receipt of a signal (action model) on an input of some unit of the object, linked to its propagation to space. Thus, the delay link can be attributed and to connection elements.
The nonstationary delay component possessing a dispersion, and its special case, unit of pure delay, also as well as the elementary inertia unit is dynamic because its output signal is peculiar, cannot be gained an inertialess composition of others, only the inertia state variables. It is result of time delay of such composition.
For generalisation of state equations of the pointwise objects presented in the form of Cauchy, on extended objects and objects with transportation delay we will formally inject the prediction operator Fwd{τ} [7]:
(2.1.1)
This operator generally, naturally, physically is not implemented, as should predict terrain clearancely precisely a value which it effects, on a finite interval of time τ forwards. But this operator is necessary only for the formal "beautiful" initial representation of state equations, and their structural solution is is possible with usage of the implemented operator of delay. On the other hand, the prediction operator in state equations acts only on a state variable such which values are determined by previous history of behaviour of all state variables of the object with delay and input actions, i.e. some composition of those, and consequently, in this special case, it is implemented, as the prognosis is rigidly determined by previous history.
So, we will record vector equations of state variables of the extended dynamic object in a kind:
(2.1.2)
In (2.1.2) as a matter of convenience writings and readings state variables are parted on two groups. Variables x(1) the first group it is state variables of the elementary inertia units of the object, their output variables. Variable x(2) it is the state variables matching to outputs of links of delay of the object. Obviously, that basically, the "inertia" and "retarding" state variables can be recorded and enumerated and in a random order and are integrated in one vector equation.
Let's mark, that the generalised set of a state equations of the dynamic object has only one explanatory variable - a time t. The space characteristics of the object in (2.1.2) are described indirectly, by the count of a vector of times of delay τ, stipulated by propagation the actions in space with final (not infinite) a velocity or transportation delay.
Consideration of dynamic objects with delay on the basis of the description their by state equations was spent by some authors and earlier [8, 10].
In [8] items 2.1, (2.1.2), the description limits the direction on delay only in right members of the equations and does not include delay links in model structure of in the capacity of operating units defined by characteristic state variables. Similar initial representation of state equations uses and in [10], «1.5. Optimum control of systems with transportation delay», p. 188 and further.
The form of the equations (2.1.2) differs from offered in [8, item 2.1, (2.1.2)] by introduction of the special state variables matching to output variables of delay links. Delay links are attributed by this with the elementary dynamic and the description of dynamic objects becomes universal.
In offered representation of the dynamic object in the present article the current inwardness of the object completely is determined by a vector of values of the state variables matching to output variables of integrators and delay links, and previous history of their behaviour.
The equations of state variables of the dynamic object with delay can be presented in integrated - "retarding" form which, perhaps, is more evident for compilation of a structural object model:
(2.1.3)
where delay operators:
(2.1.4)
execute the return on operation in relation to the prognosis operator Fwd{.}.
So, (2.1.3) - integrated - "retarding" equations of vector state variables of the multi-dimensional stretched nonlinear nonstationary dynamic object. The part of the variables which matching to output signals of the elementary inertia units and have been marked out by a vector x(1), is result of accumulation (integration) of some combination of all variables which, as well as variables, and also input actions, can vary eventually. The second part of the state variables which have been marked out x(2), represents delay of some combination of all state variables, together with object input actions, on some time τ (vector) which can vary generally eventually. According to these equations models of dynamic objects can be builted structural, including is virtual-analogue [7].
In the equations (2.1.3) initial conditions for links (operators) of delay it is not simple values of combinations of state variables and input actions in a zero instant as it occurs for integrators. For the unequivocal solution of the equations (2.1.3) it is required to set initial conditions for links of delay in the form of the functions defining history of behaviour of input values of these links on that time slice back on which they execute delay.
Thus delay links, possessing "storage", demand more information for the unequivocal solution of a problem on behaviour of the object: not simply vector of values of state variables in some, conditionally zero instant as it is enough of it for integrators, but a vector of functions (combinations of state variables and input actions on the object), set on matching to links of delay the time intervals previous the beginning of integration.
In other words, a status and behaviour of the dynamic object as points and trajectories in a state space for systems with delay it is determined not only a point position in this space, but also its previous trajectories as in a subspace of "delays" x(2), and in a subspace x(1) the "inertia" variables, and also history of behaviour of external actions during those time intervals on which there is a delay in appropriate links of delay.
Analogous statement for the traditional form of representation of state equations of objects with delay is resulted and in [8] items 2.1:
«The status of the continuous object with delay in any moment of time is characterised not only some limited number of parametres ( mean state variables - F.B.T.) (as in case of objects without delay), but also some functions determined accordingly on the interval [t0 – τe, t0], [t0 – θr, t0]. It considerably complicates control problem solving by such objects».
Generally speaking, the problem of the input of initial conditions for delay links is inherent not only to the description of the dynamic object in state variables, but also for other methods of the description. Often at digital simulation of dynamic objects with delay take an initial trajectory of "retarding" variables, i.e. output variables of delay links, a stationary value. For this purpose the link buffer is filled in a initial state in nulls or a constant.
The input signal of a delay link which is a part of the dynamic object, represents a composition of the state variables relating to other links, and actions on the object, therefore the input of the rigid prognosis of a modification of an output signal of a delay link is equivalent to the input of previous history of behaviour of the named state variables and actions for the same time interval.
Fig. 2.1.1. The status of the dynamic object with delay in some moment of time is characterised by a position of its figuring (representing) point in a state space which coordinates are values of state variables in this moment of time, and also a trajectory of this point in moments of time previous the present. It is possible to present multi-dimensional space of a status in the form of assemblage of a subspace of the inertia state variables and a subspace of the "delayed" state variables
Thus, for pointwise objects the position of a representing point in a state space in some moment of time completely determines a status of the dynamic object and a trend of its behaviour in the near future. For objects, stretched in the space, having in the structure transportation delay links, their status and the subsequent behaviour it is determined not only a current position of a representing point, but also a trajectory of its state spatial motion in previous, there can be enough major, a time interval.
Структура модели динамического объекта с запаздываниями, соответствующая системе (2.1.3) в укрупненном виде представлена на рисунке:
The structure of model of the dynamic object with the delays, matching to system (2.1.3) in the integrated kind is presented in a figure:
Fig. 2.1.2. The integrated diagrammatic representation of the main structural elements of model observable multi-dimensional nonstationary stretched in space nonlinear dynamic object of control. Own dynamic properties of the object are determined by structure, characteristics and parametres of the left block, the transformer (converter) block executes transformation of state variables to variables which can be measured (or it is direct in observed data)
Fig. 2.1.3. Structure of model of purely dynamic object, representing its interior "metabolism", i.e. transmitting directions of actions values and variables, and also the operations executed over them. The behaviour of the object with delay is determined not only a vector of initial conditions of the "inertia" state variables, but also previous history of all state variables, together with previous history of actions on the object
The complex dynamic object with delay function elements structurally is represented two parallel contours, the inertia and "retarding". State variables of all object this join of the inertia and "retarding" state variables (output variables of integrators, the elementary inertia units in object structure, and "retarding", i.e. output variables of delay links) in one vector.
As it was marked above, generally, the input signal of some link of delay is determined both all state variables of the object, and all actions on it. Therefore, to uniquely determinate a status, and then and the behaviour of the object, is necessary to know values and the prognosis of behaviour of "retarding" state variables, or, that is equivalent, previous history of behaviour of all state variables and object input actions.
As it is visible from the state equations (2.1.2) and an output equations (2.1.3) of dynamic objects with delay, for their description there are enough only four operators. The mathematical description of all four elementary units (virtual analogues of these operators) dynamic systems and the objects having the space stretch and (or) transportation delay, is mediated supporting against their physical laws describing, is reduced to the simple equations, one of which linear differential, and three remaining - algebraic:
(2.2.1)
Fig. 2.2.1. The integrator and a stationary link of delay - an exhaustive set of kinds of the elementary dynamic objects. These elementary dynamic units of objects models with delay demand for the complete and unequivocal description of a status and behaviour of the object the input of initial conditions. For the integrator it is simple value of output variable in a conditional zero moment of time, for a delay link the "initial" condition is behaviour of an input value in prior moments of time on the interval [-τ, 0], or, that the same, the prognosis of behaviour of output variable of a link of delay (a "retarding" state variable) on an interval [0, τ], equal to a time delay in a link
Fig. 2.2.2. The elementary (fundamental) units of a general kind of the block diagramme of the dynamic object as its mathematical model include only four different types of units. It is enough units of these types for simulation of as much as difficult dynamic object (the technological apparatus, it's control system, etc.)
Combining the elementary units it is possible to build well-founded model of as much as difficult dynamic object. Compilation of system of the is differential-algebraic equations of the dynamic object in the form of state equations is the implicit, mediated method, certain "sacrament", representations of model of the dynamic object in the form of a set of unidirectional elementary dynamic units co-operating among themselves.
From carrying out above consideration follows, that the unique status of the dynamic object with delay is determined not only current values of state variables, but also history of their modification in the previous moments of time, on finite and stretched enough interval. Therefore for such objects it is necessary to update concepts of observability and a controllability.
The Controllability the dynamic object with delay components consists in that there was a capability for a final time a final modification of a vector of actions to translate the object from a current status to which some certain behaviour preceded, in a new, required status to which the planned trajectory of a representing point in a state space precedes.
Observability the object with delay it is definable as a capability of determination of a current vector of state variables at any moment and a trajectory terminal phase in a state space on which the representing point comes in a current position, on measurings of output variables of the object and their behaviour during some prior time interval.
More rigorous determinations of concepts of observability and controllability of dynamic objects in representation of delays in right members of state equations can be looked in [8]: «2.6. A controllability and observability of systems with delay».
The current status of the dynamic object with delay should uniquely determinate its behaviour in the subsequent moments of time, at least on rather short interval. For lack of external actions on the object (the free motion), or at known external actions, this time is stretched ad infinitum.
The status of the dynamic object with delay is determined by an instantaneous value of all state variables, "inertia" and "retarding", and also their previous history and previous history of actions on the object.
Fig. 2.4.1. Phase portraits and behaviour of state variables of the dynamic object with delay for lack of external actions. If to consider a link with delay as elementary dynamic, i.e. to consider its output variable as a self-maintained state variable for the complete description of a status and a trend of behaviour of the dynamic object with delay it is required to set not only values of state variables in some moment of time, but also the previous history of their modification located in this case in the buffer of a link of delay. Different previous histories lead to different trajectories of a phase portrait, i.e. to different behaviour of the object. The prognosis of behaviour of output variable of a delay link (to a state variable х3) is equivalent to previous history of behaviour of its input value as represents the delay on response time, in this case τ = 1 sec, this previous history. The interval on which it is necessary to know previous history is determined by a delay factor in a delay link
Apparently, for the input of initial conditions of the state equations, and also, that is equivalent, for the unequivocal description of a current status of the dynamic object with delay it is necessary to know not only values of state variables, but also their previous history.
Fig. 2.4.2. Initial conditions or that is equivalent, a status of inertia-dynamic objects and inertia-dynamic objects with delay. For the pure inertia object for the multifold description of its properties it is enough to know values of all state variables in some moment of time, and also, values of input actions on the object if those exist. The object with delays requires not only knowledge of values of all state variables as the inertia (output signals of integrators of model), and "retarding" (output signals of delay links of model) but also to have the prognosis of behaviour of "retarding" variables
Thus, for the description of objects with delay it is required much more information, than for simple inertia objects, that complicates their analysis and optimisation.
Models of real continuous inertia dynamic objects without delay can be builted as with usage of exclusively integrators (W (s) - model), and with usage only the elementary links of delay (W (z)-model):
Fig. 2.5.1. The models of the inertia oscillating system builted on the basis of integrators and on the basis of the elementary links of delay, executing a step delay, are equivalent, as it is visible on transition functions of output variables, х1 and z1 accordingly. Naturally, the state variables of these models matching to output variables of integrators and links of a signal delay on step of simulation are different. Therefore and trajectories of representing points of different pairs of variables - different. Certainly, for model on the elementary links of delay the trajectory of a representing point "boring enough", goes on a diagonal as both variables differ on insignificant variable that is essentially important for support of a competence and representatives of model
Let's mark, that by integrators (aperiodic links) it is impossible even approximately to model links with major enough delay while any delay without problems with any exactitude is modelled by delay links on step, it is enough to select their sufficient number only.
Fig. 2.5.2. A continuous link of delay and its numeral models. The state variable which contains the exhaustive information is output variable of a delay link taking into account previous history of behaviour of its input action. Output signals of intermediate units of discrete model of a delay link can be attributed to state variables, however, as the information in them is iterated with translation, it is enough to be limited only to output variable of all link and to consider it as the elementary unitary dynamic, which status is determined not only value of output variable, but also its prognosis (previous history of an input value). The buffer of unitary discrete model is filled by previous history of an input value, therefore the state variable prognosis is rigidly determined by this previous history
Определение переменной состояния, отнесенной к звену запаздывания, собственно, равной последней величине микрозвеньев буфера запаздывания, позволяет использовать в качестве состоятельного подпространства состояний такое, которое включает только выходные величины элементарных звеньев, составляющих цифровую модель звена запаздывания. Относительно малое число эффективных переменных состояния особенно важно при аналитическом исследовании динамического объекта и графическом представлении его результатов.
Determination of the state variable attributed to a delay link, to purely, equal last variable of microlinks of the buffer of delay, allows to use in the capacity of a well-founded subspace of statuses such which includes only output variables of the elementary links making numeral model of a delay link. Rather small number of effective state variables especially important at analytical examination of the dynamic object and a graphical representation of its results.
The delay link on finite time can be considered in addition to the integrator as the elementary dynamic unit which output variable is a self-maintained state variable, and for the complete and unequivocal description of an object state it is necessary to know both a position of a representing point in a state space, and a part of its previous trajectory, т.е previous history of behaviour of the object.
The optimum control system if it is already realised, exists objectively and its characteristics do not depend on it has been described by what mathematical apparatus and with what help of mathematical methods and instruments it has been optimised. Therefore the ease of the mathematical system definition of control system , in particular the ARS (automatic regulation system), should be determined by system complexity, to it to correspond..
The author expresses gratitude to senior lecturer PhD (k.t.s.) Klinachyov Н.В (SUSU, Chelyabinsk) and to d.t.s., professor Kolesov J.B. (S-PbSTU, S-Peterburg) for the useful discussion of the problems considered in the article.
10.08.2011
* * *
