Stability of digital feedback control systems

. Specific problems arising, when Von Neumann type computer is used as feedback element, are considered. It is shown, that due to specifics of operation this element introduce pure lag into control loop, and lag time depends on complexity of algorithm of control. Method of evaluation of runtime between reading data from sensors of object under control and write out data to actuator based on the theory of semi-Markov process is proposed. Formulae for time characteristics estimation are obtained. Lag time characteristics are used for investigation of stability of linear systems. Digital PID controller is divided onto linear part, which is realized with a soft and pure lag unit, which is realized with both hardware and software. With use notions amplitude and phase margins, condition for stability of system functioning are obtained. Theoretical results are confirm with computer experiment carried out on the third-order system.


Introduction
Including Von-Neumann type computer to digital control loops born many problems, main of which is the problem of gap from emergence a situation, which requires an adequate control system response, till the real action, affected onto the controllable object [1][2][3]. Time lag depends on a number of factors, such as computer architecture, clock frequency, instructions structure, operating environment, scheduling discipline, transactions order, mathematical foundation of control algorithms, etc.
Control algorithms have next specific features, which had been studied by several authors [4][5][6]: Control algorithms have next specific features, which had been studied by several authors [4][5][6]: algorithms are a cyclic ones, i.e. they have start operator, but does not have the end operator; quest of peripherals is realized by means of inclusion into algorithm special transaction management operators; for an external observer selection of branch in places of algorithm ramification is a stochastic one, and probabilities of branching depend on a distribution of data processed; for an external observer algorithm operators' run-time is, a random one, distribution function of time of operator execution depends on distribution of data processed.
Thus, a process of a deterministic algorithm interpretation with Von-Neumann type controller for external observer by several authors [5,7,8] is regarded as semi-Markov process with continuous time. Operators of an algorithm are considered as states of semi-Markov process. Interpretation of algorithm may be considered as sequence of state switches or wandering through the states of semi-Markov process.
Among the algorithm operators there are operators, which request sensors and operators, which request actuator of the object under control. Time intervals between semi-Markov process states, which are abstract analogues of operators mentioned and influence of intervals onto the quality of control in linear control systems is subject of following investigations.

General semi-Markov model of the control algorithm
The flowchart of digital feedback control system is shown on the Figure 1  The system includes object under control and the Von-Neumann type computer as a controller. The physical state of object under control is estimated by sensor, which generates data for computer processing. Output of Von-Neumann type computer through actuator affects on the object, so the feedback is closed. Model of a system soft is the semi-Markov process [8][9][10][11]  ( ) -is the pure time density of residence in the state j a with further switch into the state n a .
Common structure of semi-Markov process, which simulates the soft of computer, is shown on the Figure 2 a. The process is ergodic, and is performed by the full graph with loops. From this structure all possible structures with J states may be obtained. In context of the task under solution the set of states A may be divided onto three disjoint subsets (classes): To avoid second case one should to simplify the semi-Markov process (1) with use method, described in [12], as follows: In turn, simplified process (3) may be transformed into the process μ ′ ′ , shown on the Figure 2 b as follows: one should to evaluate probabilities of residence of semi-Markov process (4) in states of set (5) as follows [5,7,13]: where m T is the time of a residence the process (4) in the state m a ; m τ is the time of a return the process (4) at the state m a ; ( )

Stability of linear systems
Let us consider an influence of lag time ( ) t f sa on the stability of digital feedback control system, shown on the Figure 1 for the case, when model of object under control is the linear one. Control action is calculated as linear combination of sensor data, derivatives of the first and higher orders and integrals of the first and higher orders of sensor data (PID controller). Structural scheme of the linear system is shown on the Figure 3 [1,2]. ( ) λ cd W is the transfer function of computer λ is the Laplace variable (the differentiation operator); sa t is a random parameter, in accordance with section 2.
Substitution into transfer function where ω is the circular frequency; gives next form of the transfer function from the input of object under control till the output of computer without taking into account pure lag: where ( ) ( ) is the phasefrequency response. Both amplitude-frequency response and phasefrequency response are shown on the give for a system without lag amplitude and phase margins respectively [1,2].
Let us insert the unit ( ) λ − sa t exp into the control loop. The amplitude response of the unit is equal to 1 in all frequency range [16,17,18,19]. Phase response has the form of an inclined straight line with the slope factor equal to sa t (shown on the Figure 4 with dashed line).
In ( ) ( ) ω Summing the values of straight line with values of the curve increases a steepness of the curve. Increase of a steepness leads to decrease of amplitude g α Δ and phase g ϕ Δ margins. If margins became negative the system losses stability. So, to provide stability of the system one should to operate with use next simple method.
Method of a digital feedback control system stability estimation.
1) For the system ( ) λ W define amplitude and phase margins g α Δ and g ϕ Δ .
2) For the control algorithm define max as T . 3) Check the next criteria for the system with a lag: If both conditions are satisfied, the system is stable, if not, there is necessary to change system hardware or software.

Example
As an example one would consider the system, in which linear part is performed by the transfer function System response on the Heavisaid function is shown on the Figure 5. . This is coтfirmed by the graphs of transition processes. Process diverges when 5 , 1 = sa t , and converges in all other cases.

Conclusion
Von-Neumann type computer lag time is the important factor, which should be taken into account when working out a digital control system, both hardware and software. Working out approach to evaluation a behavior of the system with pre-determined configuration opens new page in the theory and practice of system design. With use results expounded above one can to establish demands to software, especially to its time characteristics. This, in turn permits to limit computational complexity of control algorithm and facilitate decision making by software developers.
Further investigation in this area should be directed to working out practical recommendations for working out algorithm satisfied obtained limitations, and to finding links with other methods of software design.