Control Systems and Adaptive Process. Design, and control methods and strategies 1.

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Control Systems and Adaptive Process. Design, and control

methods and strategies

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Controllability and observability Controllability

• Consider the system of n states and p inputs

with constant matrices and . The states equation, or the pair (A, B), is said to be controllable if for any initial state and any final state , there is an input that transfers the state x from x0 to x1 in finite time. Otherwise, the equation (1.1), or the pair (A, B), is said non-controllable.

BuAxx

nxnA nxpB

nxx 0)0( nx 1

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• You can determine if the system is controllable by examining the algebraic condition:

• Matrix A has dimension n x n and B n x 1. For systems with multiple input matrix B is n × m, where m is the number of inputs.

• For a system of single-input single-output, controllability matrix Pc is described in terms of A and B as:

which is an n x n matrix, therefore, if the determinant of Pc is not zero, the system is controllable.

nBABAABBrank n 12

BABAABBP nc

12

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• Example: consider the system

from which we have that

The determinant of Pc = 1 ≠ 0, so that the system is controllable.

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aaa

tx

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1

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)(1

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aaa

aBAABBPc

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Controllability and observability Controllability test

• The following statements are equivalent:1. The pair , is controllable2. The controllability matrix

is of rank n (full row rank). 3. Matrix n x n

is nonsingular for all t > 0.

pnnn BABA ,),,(

npnn CBABAABBC 12

t

tATtAt

ATAc deBBedeBBetW

TT

0

)()(

0

)(

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Controllability and observability Minimum energy control

• Control spending minimum energy to bring the system from state x0 to state x1 at time t1, in the sense that, for other control ũ(t) to make the same transfer, is always true that:

It is observed that the minimum control power is greater when the distance between x0 and x1 is greater, and the transfer time t1 is lower x0 y x1.

2

10)(

12

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10)(

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11

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))(())(()(

))(()()()()(~

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xxetWxxetWxex

xetWdeBBetWxexdudu

tAc

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xtAc

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tATtAc

Tt

tATt

T

TT

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Controllability and observability Controllability PBH tests

• The Popov-Belevitch Hautaus (PBH) tests have interesting geometric interpretations used to analyze the controllability in the form of Jordan. There are two types of test, of eigenvectors and of rank.1. Eigenvectors test: The pair (A,B) is not controllable if and only if there is a left eigenvector of A such that

2. Rank PBH test: Pair (A,B) is controllable if and only if

nv 1

0vB

severyfornBAsIrank

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Controllability and observability Controllability PBH tests

• Controllability and similarity transformation: invariance theorem regarding controllability coordinate changes..

Controllability is an invariant property with respect to equivalence transformations (coordinate changes).

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Controllability and observability Observability

• All poles of a closed-loop system can be placed arbitrarily in the complex plane if and only if the system is observable and controllable. Observability refers to the possibility of estimating a state variable.

• According to R. Dorf, a system is completely observable if and only if there exists a finite time T such that the initial state x(0) can be determined from the observation of history y(t) given the control u(t).

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• Considering the system of one input and one output

where C is a row vector 1 x n and x is a column vector n x 1. This system is fully observable when the determinant of the observability matrix Po is nonzero, where

which is a matrix of n x n.

CxyandBuAxx

1n

o

CA

CA

C

P

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• Example: Consider the system

Therefore and Is thus obtained

the determinant of Po = 1 and the system is fully observable. Note that the determination of the observability matrix does not use matrices B and D.

210

100

010

aaa

A 001C

010CA 1002 CA

100

010

001

oP

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• The concept of observability is dual to controllability. Tries to find out the possibility of estimating the system state from the knowledge of the output. Consider the steady linear system

This state equation (1.2) is observable if for any unknown initial state x(0), there is a finite time t1 such that the knowledge of the input u and the output y on the interval [0,t1] is sufficient to determine uniquely the initial state x(0). Otherwise the system is not observable.

BuAxx

DuCxy

pqnqpnnn DCBA ;;;

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Controllability and observability State variables

• For a given system, they exist plenty of possible sets of state variables. However, all possible sets must consist of the same number of state variables and the defined variables must be fully independent. Understanding as independent variable that whose value cannot be expressed in terms of the other variables; which implies that the initial values of each of the chosen state variables may be assigned freely.

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• Example, in a system such as shown in figure 3.1 may be taken as state variables the speed ẏ(t) of the mass M and the force ky(t) in the spring; the strength in the spring and the displacement y(t) of the mass may not be taken, since the former is equal to the second multiplied by the constant K. Another valid alternative would be to take as state variables of the system the speed ẏ(t) and the displacement y(t) of the mass.

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• General methods for the selection of the state variables of a system:

- Method of physical variables: the selection of the state variables is performed based on the energy storage elements existing in the system.

- Method of phase variables.- Jordan canonical form.

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• Linear systems with variable parameters: in a system whose dynamic behavior is characterized by

This equation can be represented by the following state and output equations

Calculating coefficients Bi(t) by means of

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• Obtaining the transfer function from the state equations:– The transfer matrix or function of a linear time-invariant

system can be obtained from the state equations of the system by applying the Laplace transform.

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Bibliography

• R. Dorf, R. Bishop: Modern control systems.• Class notes ETSII. UNED

Interesting links• http://iaci.unq.edu.ar/materias/control2/web/clases/Cap6.pdf • http://www.slideshare.net/IsRrItA/variables-de-estado • http://www.virtual.unal.edu.co/cursos/ingenieria/2001619/lecciones/

estado/node4.html#SECTION00631000000000000000

Control Systems and Adaptive Process. Design, and control methods and strategies 1.

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