Controllability

A system is fully controllable if a state can be converted to any new state in a finite time using suitable control signals. A system can be controlled if it can be transferred from selected initial states to selected final states. The controllability thus describes the influence of external input variables (usually the control variables) on the internal system state. A distinction is made between output controllability and state controllability .

Colloquially, the term controllable is often used in the sense of controllable . In technical terms, however, a distinction is made between controllable , observable and controllable . In order for a system to be controllable , it must be possible both to observe its state and to control it . Controllability is usually relevant for practical application . Controllability is one aspect of this.

The concept of controllability and observability was introduced after 1960 by Rudolf Kálmán .

definition

The starting point for assessing the controllability of a linear system is the state space representation

${\displaystyle \mathbf {\dot {x}} =\mathbf {A} \cdot \mathbf {x} +\mathbf {B} \cdot \mathbf {u} }$
${\displaystyle \mathbf {y} =\mathbf {C} \cdot \mathbf {x} +\mathbf {D} \cdot \mathbf {u} }$

with the system matrix , the control matrix , the observation matrix , the passage matrix , the state vector , the output vector and the control vector . ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} }$ ${\displaystyle \mathbf {C} }$ ${\displaystyle \mathbf {D} }$ ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {y} }$ ${\displaystyle \mathbf {u} }$

To determine controllability, there are various criteria that depend on the form of the state space representation.

Complete controllability

A linear system is called fully state controllable (sometimes also called achievable) if there is a control function for each initial state that converts the system to any final state within any finite period of time . ${\displaystyle \mathbf {x} (t_{0})}$ ${\displaystyle \mathbf {u} (t)}$ ${\displaystyle t_{0}\leq t\leq t_{e}}$ ${\displaystyle \mathbf {x} (t_{e})}$

Structural controllability

A class of systems is called structurally controllable if there is at least one system that is completely controllable. ${\displaystyle S\ (S_{A},\ S_{B},\ S_{C})}$ ${\displaystyle (A,B,C)\in S}$

Thereby are matrices in which all elements not equal to 0 have been marked with *, since all elements equal to 0 determine the structural observability and structural controllability. I.e. the det S must not be equal to 0. ${\displaystyle S\ (S_{A},\ S_{B},\ S_{C})}$

Controllability Criteria

Complete output controllability

The system is fully output controllable if and only if the rank of the matrix ${\displaystyle (\mathbf {A} ,\mathbf {B} ,\mathbf {C} ,\mathbf {D} )}$

${\displaystyle \mathbf {S} _{AS}={\begin{pmatrix}\mathbf {CB} &\mathbf {CAB} &\mathbf {CA^{2}B} &\cdots &\mathbf {CA^{n-1}B} &\mathbf {D} \end{pmatrix}}}$

corresponds to the number of output variables: The condition for output controllability is therefore rank . Provided that rank (C) = r, every state-controllable system is also output-controllable. The reverse is not true. ${\displaystyle \mathbf {S} _{AS}=r}$

Complete state controllability

Kalman's criterion

The system is fully controllable according to Kalman if and only for the controllability matrix ${\displaystyle (\mathbf {A} ,\mathbf {B} )}$

${\displaystyle \mathbf {S} _{S}={\begin{pmatrix}\mathbf {B} &\mathbf {AB} &\mathbf {A^{2}B} &\cdots &\mathbf {A^{n-1}B} \end{pmatrix}}}$

applies

${\displaystyle \mathrm {Rang} \ \mathbf {S} _{S}{}={}n}$.

In special cases , controllable systems can even be inverted, which is a prerequisite for using Ackermann's formula for setting poles for single-variable systems . The Kalman state controllability is a special case of the complete output controllability for . ${\displaystyle \mathbf {B} \in \mathbb {R} ^{n\times 1}}$ ${\displaystyle \mathbf {S} _{S}}$ ${\displaystyle \mathbf {C} =\mathbf {I} ,\ \mathbf {D} =\mathbf {0} }$

Gilbert's criterion

The system , whose state space model is in canonical normal form, is fully controllable according to Gilbert if the matrix has no zero line and if the p lines of the matrix , which belong to the canonical state variables of a p-fold eigenvalue, are linearly independent. ${\displaystyle ({\rm {diag}}\lambda _{i},{\tilde {\mathbf {B} }})}$ ${\displaystyle {\tilde {\mathbf {B} }}}$ ${\displaystyle {\tilde {\mathbf {b} }}_{i}^{T}}$ ${\displaystyle {\tilde {\mathbf {B} }}}$

${\displaystyle {\frac {{\rm {d}}{\tilde {\mathbf {x} }}(t)}{{\rm {d}}t}}={\rm {diag}}\lambda _{i}\cdot {\tilde {\mathbf {x} }}(t)+{\tilde {\mathbf {B} }}u(t),\ {\tilde {\mathbf {x} }}(0)=\mathbf {V} ^{-1}\mathbf {x} _{0}}$

${\displaystyle \mathbf {V} }$is the matrix of the eigenvectors of the matrix belonging to the eigenvalues , with . ${\displaystyle \mathbf {A} }$ ${\displaystyle {\tilde {x}}=\mathbf {V} ^{-1}x}$

Hautus' criterion

The system (A, B) is completely controllable according to Hautus if and only if the condition

${\displaystyle \mathrm {Rang} {\begin{pmatrix}\lambda _{k}I-\mathbf {A} &|&\mathbf {B} \end{pmatrix}}=n}$

for all eigenvalues , the matrix A is fulfilled. ${\displaystyle \lambda _{k}}$ ${\displaystyle k=1,2,\ldots ,n}$

Controllability of scanning systems

The above relationships also apply to scanning systems when replaced by the transition matrix and by the discrete input matrix. According to, the check can be simplified by checking first the conditions for the continuous system and then the additional condition ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} }$ ${\displaystyle \mathbf {H} }$

${\displaystyle e^{s_{i}T_{ab}}\neq e^{s_{j}T_{ab}}}$ For ${\displaystyle s_{i}\neq s_{j}}$

is satisfied.

Normal control form (Frobenius form)

The normal control form can be easily determined from the transfer function: The following applies to: ${\displaystyle G(s)={\frac {b_{0}+b_{1}s+\dotsb +b_{n-1}s^{n-1}+b_{n}s^{n}}{a_{0}+a_{1}s+\dotsb +a_{n-1}s^{n-1}+a_{n}s^{n}}}}$ ${\displaystyle a_{n}=1}$

Block diagram of state space representation

${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\...\\{\dot {x}}_{n-1}\\{\dot {x}}_{n}\\\end{bmatrix}}=\underbrace {\begin{pmatrix}0&1&0&...&0\\0&0&1&...&0\\0&0&0&...&0\\...&...&...&...&...\\0&0&0&0&1\\-a_{0}&-a_{1}&-a_{2}&...&-a_{n-1}\\\end{pmatrix}} _{A_{R}}{\begin{bmatrix}x_{1}\\x_{2}\\...\\x_{n-1}\\x_{n}\\\end{bmatrix}}+\underbrace {\begin{bmatrix}0\\0\\...\\1\\\end{bmatrix}} _{b_{R}}u}$

${\displaystyle y=\underbrace {\begin{bmatrix}b_{0}-b_{n}a_{0}&b_{1}-b_{n}a_{1}&\cdots &b_{n-1}-b_{n}a_{n-1}\\\end{bmatrix}} _{c'_{R}}{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n-1}\\x_{n}\\\end{bmatrix}}+{\begin{matrix}\underbrace {b_{n}} \\{\textrm {}}^{\rm {d_{R}}}\end{matrix}}u}$

or for systems without derivatives of the input variable

${\displaystyle y=\underbrace {\begin{bmatrix}b_{0}&0&\cdots &0\\\end{bmatrix}} _{c'_{R}}{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n-1}\\x_{n}\\\end{bmatrix}}+0u}$

The special form of and is helpful for the analysis and construction of state controllers . ${\displaystyle A_{R}}$ ${\displaystyle b_{R}}$

Nonlinear controllability and flatness

In the non-linear, you cannot make any global statements about controllability and must always link this to a validity range. The mathematical ad operator plays a special role here.

Therefore, the system property of flatness extends the controllability to the nonlinear case. In the linear case, controllable systems are also flat.

However, caution is required when inferring the controllability of the nonlinear system from the linearization. If the linearization can be controlled around a point, the non-linear system can be controlled locally around this point. However, if the linearization is not controllable, the system can still be controllable.

Reasons for not fully controllable systems

There are two main reasons for the incomplete controllability:

1. Internal processes that are not connected to the input cannot be controlled.
2. Two parallel subsystems with the same dynamic properties cannot be fully controlled.

Reasons for investigation

The controllability criterion can also be used to simplify a control task. If it is not the manipulated variable but rather the disturbance variable that is examined for its controllability with regard to the controlled variable, non-controllability shows that this part of the system is not subject to the influence of the disturbance and that this part does not have to be regulated if the disturbance is to be suppressed. On the other hand, a disturbance variable cannot be compensated if a system part can be controlled by the disturbance variable but not by the manipulated variable.

The property that the disturbance cannot be controlled is used for some control methods. In the case of interference decoupling, the controller is developed in such a way that the manipulated variable no longer depends on the disturbance variable.

See also

• State space representation
• Control engineering
• Observability

Individual evidence

1. ↑ Otto Föllinger: Control engineering, introduction to the methods and their application . 8th edition. Hüthig Verlag, Heidelberg 1994, ISBN 3-7785-2336-8 . , Section 12.3.1
2. ↑ Lunze, Jan: Control engineering 2: multi-variable systems digital control. S. 84, 4th edition Heidelberg: Springer, 2006. - ISBN 3-540-32335-X
3. ↑ Lunze, Jan: Control engineering 2: multi-variable systems digital control. P. 64, 4th edition Heidelberg: Springer, 2006. - ISBN 3-540-32335-X
4. ↑ Lunze, Jan: Control engineering 2: multi-variable systems digital control. P. 73, 4th edition Heidelberg: Springer, 2006. - ISBN 3-540-32335-X
5. ↑ Lunze, Jan: Control engineering 2: multi-variable systems digital control. S. 75, 4th edition Heidelberg: Springer, 2006. - ISBN 3-540-32335-X
6. ↑ Jürgen Ackermann: Sampling control; 1. Analysis and synthesis . 2nd Edition. Springer, Heidelberg 1983.
7. ↑ Lunze, Jan: Control engineering 2: multi-variable systems digital control. P. 76, 4th edition Heidelberg: Springer, 2006. - ISBN 3-540-32335-X