Legendre polynomials

Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.

Controllability and observability are dual aspects of the same problem.

Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.

The following are examples of variations of controllability notions which have been introduced in the systems and control literature:

• State controllability
• Output controllability
• Controllability in the behavioural framework

State controllability

The state of a system, which is a collection of the system's variables values, completely describes the system at any given time. In particular, no information on the past of a system will help in predicting the future, if the states at the present time are known.

Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input to move the internal state of a system from any initial state to any other final state in a finite time interval.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

Continuous linear systems

Consider the continuous linear time-variant system

$\dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t)$

$\mathbf{y}(t)=C(t)\mathbf{x}(t)+D(t)\mathbf{u}(t).$

There exists a control $u$ from state $x_0$ at time $t_0$ to state $x_1$ at time $t_1>t_0$ if and only if $x_1-\varphi (t_0,t_1)x_0$ is in the column space of

$W(t_0,t_1)=\underset{t_0}{\overset{t_1}{\int }}\varphi (t_0,t)B(t)B(t{)}^T\varphi (t_0,t{)}^{T}dt$

where $\varphi$ is the state-transition matrix.

In fact, if ${\eta }_0$ is a solution to $W(t_0,t_1)\eta =x_1-\varphi (t_0,t_1)x_0$ then a control given by $u(t)=-B(t{)}^T\varphi (t_0,t{)}^T{\eta }_0$ would make the desired transfer.

Note that the matrix $W$ defined as above has the following properties:

• $W(t_0,t_1)$ is symmetric
• $W(t_0,t_1)$ is positive semidefinite for $t_1\ge t_0$
• $W(t_0,t_1)$ satisfies the linear matrix differential equation

$\frac{d}{dt}W(t,t_1)=A(t)W(t,t_1)+W(t,t_1)A(t{)}^T-B(t)B(t{)}^T,W(t_1,t_1)=0$

• $W(t_0,t_1)$ satisfies the equation

$W(t_0,t_1)=W(t_0,t)+\varphi (t_0,t)W(t,t_1)\varphi (t_0,t{)}^T$

Continuous linear time-invariant (LTI) systems

Consider the continuous linear time-invariant system

$\dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)$

$\mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)$

where

$\mathbf{x}$ is the $n\times 1$ "state vector",

$\mathbf{y}$ is the $m\times 1$ "output vector",

$\mathbf{u}$ is the $r\times 1$ "input (or control) vector",

$A$ is the $n\times n$ "state matrix",

$B$ is the $n\times r$ "input matrix",

$C$ is the $m\times n$ "output matrix",

$D$ is the $m\times r$ "feedthrough (or feedforward) matrix".

The $n\times nr$ controllability matrix is given by

$R=\left[\begin{array}{ccccc}B& AB& A^{2}B& ...& A^{n-1}B\end{array}\right]$

The system is controllable if the controllability matrix has full rank (i.e. $\mathrm{rank}(R)=n$).

Discrete linear time-invariant (LTI) systems

For a discrete-time linear state-space system (i.e. time variable $k\in \mathbb{\mathbb{Z}}$) the state equation is

$\mathbf{\text{x}}(k+1)=A\mathbf{\text{x}}(k)+B\mathbf{\text{u}}(k)$

Where $A$ is an $n\times n$ matrix and $B$ is a $n\times r$ matrix (i.e. $\mathbf{u}$ is $r$ inputs collected in a $r\times 1$ vector. The test for controllability is that the $n\times nr$ matrix

$\mathcal{𝒞}=\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]$

has full row rank (i.e., $\mathrm{rank}(C)=n$). That is, if the system is controllable, $C$ will have $n$ columns that are linearly independent; if $n$ columns of $C$ are linearly independent, each of the $n$ states is reachable giving the system proper inputs through the variable $u(k)$.

Example

For example, consider the case when $n=2$ and $r=1$ (i.e. only one control input). Thus, $B$ and $AB$ are $n\times 1$ vectors. If $\left[\begin{array}{cc}B& AB\end{array}\right]$ has rank 2 (full rank), and so $B$ and $AB$ are linearly independent and span the entire plane. If the rank is 1, then $B$ and $AB$ are collinear and do not span the plane.

Assume that the initial state is zero.

At time $k=0$: $x(1)=A\mathbf{\text{x}}(0)+B\mathbf{\text{u}}(0)=B\mathbf{\text{u}}(0)$

At time $k=1$: $x(2)=A\mathbf{\text{x}}(1)+B\mathbf{\text{u}}(1)=AB\mathbf{\text{u}}(0)+B\mathbf{\text{u}}(1)$

At time $k=0$ all of the reachable states are on the line formed by the vector $B$. At time $k=1$ all of the reachable states are linear combinations of $AB$ and $B$. If the system is controllable then these two vectors can span the entire plane and can be done so for time $k=2$. The assumption made that the initial state is zero is merely for convenience. Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).

This example holds for all positive $n$, but the case of $n=2$ is easier to visualize.

Analogy for example of n = 2

Consider an analogy to the previous example system. You are sitting in your car on an infinite, flat plane and facing north. The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line. If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north). The lack of steering case would be analogous to when the rank of $C$ is 1 (the two distances you drove are on the same line).

Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of $C$ is 2.

If you change this example to $n=3$ then the analogy would be flying in space to reach any position in 3D space (ignoring the orientation of the aircraft). You are allowed to:

• fly in a straight line
• turn left or right by any amount (Yaw)
• direct the plane upwards or downwards by any amount (Pitch)

Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.

Nonlinear systems

Nonlinear systems in the control-affine form

$\dot{\mathbf{x}}=\mathbf{f}\mathbf{(}\mathbf{x}\mathbf{)}+\sum _{i=1}^m{\mathbf{g}}_i(\mathbf{x})u_i$

is locally accessible about $x_0$ if the accessibility distribution $R$ spans $n$ space, when $n$ equals the rank of $x$ and R is given by:

$R=\left[\begin{array}{cccccc}{\mathbf{g}}_1& \cdots & {\mathbf{g}}_m& [{\mathrm{a}\mathrm{d}}_{{\mathbf{g}}_i}^k{\mathbf{g}}_{\mathbf{j}}]& \cdots & [{\mathrm{a}\mathrm{d}}_{\mathbf{f}}^k{\mathbf{g}}_{\mathbf{i}}]\end{array}\right].$

Here, $[{\mathrm{a}\mathrm{d}}_{\mathbf{f}}^k\mathbf{g}]$ is the repeated Lie bracket operation defined by

$[{\mathrm{a}\mathrm{d}}_{\mathbf{f}}^k\mathbf{g}]=\left[\begin{array}{ccccc}\mathbf{f}& \cdots & j& \cdots & \mathbf{[}\mathbf{f},\mathbf{g}\mathbf{]}\end{array}\right].$

The controllability matrix for linear systems in the previous section can in fact be derived from this equation.

Output controllability

Output controllability is the related notion for the output of the system; the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular:

• A controllable system is not necessarily output controllable. For example, if matrix D = 0 and matrix C does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states. A trivial numerical example uses D=0 and a C matrix with at least one row of zeros; thus, the system is not able to produce a non-zero output along that dimension.
• An output controllable system is not necessarily state controllable. For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant zero dynamics, which are trajectories of the system that are not observable from the output. Consequently, being able to drive an output to a particular position in finite time says nothing about the state configuration of the system.

For a linear continuous-time system, like the example above, described by matrices $A$, $B$, $C$, and $D$, the $m\times (n+1)r$ output controllability matrix

$\left[\begin{array}{cccccc}CB& CAB& C{A}^{2}B& \cdots & C{A}^{n-1}B& D\end{array}\right]$

must have full row rank (i.e. rank $m$) if and only if the system is output controllable.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger This result is known as Kalman's criteria of controllability.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Controllability under input constraints

In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns). The controllability of systems with input and state constraints is studied in the context of reachability and viability theory.

Controllability in the behavioural framework

In the so-called behavioral system theoretic approach due to Willems (see people in systems and control), models considered do not directly define an input–output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.

A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external veriables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.

To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for

One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier

The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved

First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen

The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01

Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

Stabilizability

A slightly weaker notion than controllability is that of stabilizability. A system is determined to be stabilizable when all uncontrollable states have stable dynamics. Thus, even though some of the states cannot be controlled (as determined by the controllability test above) all the states will still remain bounded during the system's behavior.

See also

• Observability
• State observer

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Template:Planetmath reference
• MATLAB function for checking controllability of a system
• Mathematica function for checking controllability of a system