Systematicstate variableA performance that can be controlled by external input.If within a limited time interval, you can useamplitudeThere is no limited input effect, making the deviation from the systemEquilibrium stateWhen an initial state of is returned to the equilibrium state, it is said that the initial state is controllable.
When all possible initial states of the system are controllable, the system is said to be fully controllable; otherwise, the system is said to be incompletely controllable.[1]The concept of controllability is defined by RE. Kalman first put forward it in 1960, and it soon becamemodern control theory A fundamental concept in solving thepole assignment 、optimum controlAnd so on.For linear systems（state variableAnd output variables are satisfied for all possible input variables and initial statessuperposition principle There are mature research results on controllability and its criteria.From the perspective of control system design, it is possible to design appropriatestate feedback sendClosed loop control systemIt has arbitrarily specified performance.If only the closed-loop control system designed is asymptotically stable (seeMotion stability）Then the fully controllable condition can be relaxed to be incompletely controllable, and the uncontrollable part is required to be stable.
Controllability and observability
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Controllability andObservabilityIt is a relative concept.
dynamical system Controllability and observability are two important basic structural characteristics that reveal the invariant nature of dynamic systems.[2]
Kalman first proposed state controllability and observability in the early 1960s.Subsequent development shows that these two concepts can answer basic questions such as whether the controlled system can be controlled and integratedstate estimation The study of the problem is of great significance.
System controllability refers to the possibility that the control action controls the state and output of the controlled system.
The observability reflection is determined by the measurement value of the input and output that can be directly measuredDynamic characteristicsThe possibility of the state of.
Composed according to certain rulesBlock matrix, the expression is
expression
For the systemdimension。 There are other forms of criteria for judging controllability of linear time invariant systems.For linearTime-varying systemThe conditions for judging controllability are more complex, and whether the system can be controlled often depends on the selection of initial time.
The coefficient of the characteristic multinomial of.aboutMultivariable systemThe controllable normal form of the equation of state is more complicated in form and not unique.The commonly used ones are the Luenberg canonical form, the Wanham canonical form and the Hengshan canonical form.Controllable normal form is often used in the synthesis of control systems according to expected poles (seepole assignment ）。
less thann, the decomposedstate equation It has the following forms:
expression
Where
Fractal state
It is the controllable sub state,
Fractal state
It is unable to control the score.Subsystem
It is the uncontrollable part of the system, subsystem
It is the controllable part of the system.External input function
Can only affect controllable sub state
, without affecting the status of uncontrollable points
。From the perspective of control system design, it is possible to design appropriatestate feedback EnvoyClosed loop control systemIt has arbitrarily specified performance.However, if only the closed-loop control system designed is required to be asymptotically stable (see stability), then the fully controllable condition can be relaxed to be partially controllable, and only the uncontrollable part is required to be stable.Generally, the partially controllable system whose uncontrollable part is stable is called stable system.
aboutDistributed parameter systemandnonlinear system Controllability and its judging conditions have also been studied, but its complexity has greatly increased, and many problems remain to be solved.
The system can be controlled if it reaches full rank.
twoHautus criterion
First of all, we need to know the theorem: linear time invariant system passes through nonsingularlinear transformation The controllability of the rear state remains unchanged.
The following formula holds for all complex numbers λ
