AUTOMATIC CONTROL AND SYSTEM THEORY - … · AUTOMATIC CONTROL AND SYSTEM THEORY ... • Physical properties are usually distributed in the systems: ... approximations that allow

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C. Melchiorri (DEI) Automatic Control & System Theory 1

AUTOMATIC CONTROL AND SYSTEM THEORY

SYSTEMS AND MODELS

Claudio Melchiorri

Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione (DEI) Università di Bologna

Email: [email protected]

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Systems and Models

The terms: •  “System” •  “System theory” •  “System engineering” • …

are frequently used in many –different– areas: process control, data elaboration, biology, economy, ecology, management, traffic control, … System: common “element” in this terminology Need of defining and analysing their structural properties

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Systems and Models

System: •  A set, artificially isolated from the context, possibly made by more

(internal) interacting parts, •  It is desired to study its “dynamic” behaviour

2

1 3

4

1 – 4: connections

External environment

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Systems and Models

•  System: The time-evolution of a system is observable by means of a set of measurable attributes or variables that change in time

•  Measurable attribute: Feature of the system that can be related to one or more integer, real or complex numbers, or in any case to a set of symbols

•  Mathematical model: Expression by mathematical equations of the relationship between the measurable attributes

2

1 3

4

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Systems and Models

• The study of a system and of its properties is based on a mathematical model (although other models can be used as well) that describes, with a given approximation, the relationships among the system variables.

• Different models can be associated with a given system, and the chosen model depends on the desired representation level, precision, and complexity.

• The goal of the System Theory is to mathematically describe/represent a system in order to •  understand its main physical properties and •  design a proper control system.

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Systems and Models

• A system is graphically represented with a block, •  Its variables are indicated by links to the external environment

or other systems.

S

S1 S2

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Systems and Models

•  In an oriented system, it is possible to consider •  Input variables (causes) •  Output variables (effects)

• The separation between input and output variables is neither unique

nor clear

S

u1(t)

u2(t)

u3(t)

y(t)

input

output

Ra La c(t), ω(t)

Le va(t)

ia(t)

ve(t)

ie(t)

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Systems and Models

Distributed Parameters

Lumped Parameters

Stochastic Deterministic

Continuous time

Discrete time

Nonlinear Linear

Time Varying

Constant coefficients

Non homogeneous Homogeneous

Systems

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Systems and Models

Distributed Parameters

Lumped Parameters

Stochastic Deterministic

Continuous time

Discrete time

Nonlinear Linear

Time Varying

Constant coefficients

Non homogeneous Homogeneous

Systems

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Systems and Models: Static and Dynamic Systems

Two types of systems are considered: 1.  Static systems (memory-less)

Mathematical model of static systems: •  Algebraic equations – in a given instant, the output depends only

on the input value in that instant (e.g. the relation between the tension and the current in a resistance)

2.  Dynamic systems (with memory) Mathematical model of dynamic systems (lumped parameters): •  Differential equations – in a given instant, the output does not

depends only on the value of the input variables in that instant, but also on the previous values (e.g. the relation between the tension and the current in a capacitor)

Concept of state

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Lumped parameters models

•  Physical properties are usually distributed in the systems: •  Mass •  Elasticity •  Resistance •  ...

•  In their mathematical description, when possible it is better to introduce

approximations that allow to concentrate in one (or few) points those properties: Relevant simplifications can then be achieved in the mathematical descriptions.

•  Lumped parameters models are obtained.

•  In the practice, although it is evident that all the physical properties of systems are distributed, whenever possible it is advisable to use lumped parameter models.

Concentrated elasticity

Concentrated mass

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Lumped parameters models

Lumped parameters models are expressed by Ordinary Differential Equations, ODE, (continuous time) or Difference Equation (discrete time), that are function of time only:

When it is not possible to consider as concentrated (some of) the parameters of the system, it is necessary to use Partial Difference Equations, PDE. In this case, the dynamics is not only a function of time, but ALSO e.g. of space:

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Systems and Models

• Static system (algebraic)

• Dynamic system

i(t)

v(t) R

i(t)

v(t) R

C

0 50 100 150 200 250 300 0 10 20 30 40 50 60

Time (s)

V, I

0 0.5 1 1.5 2 0

0.2

0.4

0.6

0.8

1

q

v

v(t) = R i(t)

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Systems and Models – Static systems

• Electric circuit with resistances

• Combinational circuit

vi(t)

R1

R2 vu(t)

u1

u2 u3

y

u1 u2 u3 y

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 1

The output is a pure function of the present input only, in contrast to sequential logic in which the output depends not only on the present input but also on the history of the input. The inputs could e.g. represent the positions of switches and the output the lighting of a bulb.

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Systems and Models – Dynamic systems

• Dynamic systems: with memory. The output values in a given instant of time depend on current and previous history of inputs (history = memory).

• Electrical circuit with elements able to store energy (capacitors and/or inductors)

R1

u(t) R2 vu(t) vc(t) x

u(t) vu(t)

x

input output

state

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Systems and Models – Dynamic systems

• Electrical circuit with elements able to store energy (capacitors and/or inductors)

• One obtains

R1


u(t) vu(t)

x

input output

state

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Systems and Models

•  The description of the system is made with two equations, a differential and an algebraic one:

u x y

State differential equation

Output equation

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Systems and Models

•  The description of the system is made with two equations, a differential and an algebraic one:

•  In general, it is difficult to solve the state differential equation. However, for a given class of systems (i.e. linear, continuous time, lumped systems) and given the initial state x(0) = x0 (value of the state at t = 0), one obtains:

State transition equation

State differential equation

Output equation

Similar equations hold also in the discrete time case.

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Systems and Models

• The equation valid also for the vector case, is the so-called Lagrange’s formula

•  It is computed from the derivative of an integral (fundamental theorem of calculus)

with

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Systems and Models

Main analysis problems in System Theory are:

•  Motion analysis – Output analysis: computation of the motion of the state (time evolution of x) or of the output function y(t), given the initial state x(0) = x0 and the input function u(t)

•  Controllability analysis: how to affect the state motion or the output function acting on the input variable

•  Observability analysis: computation of the state of the system in a given instant t, given the input and output values over the course of time period

•  Sensitivity analysis: study of the influence on the state/output evolution of changes in the initial state, of the input function, of the system’s parameters

•  Stability analysis: in a stable system, limited variations of the initial state or of the input function generate limited variations of the state evolution or of the output

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Systems and Models

Main synthesis problems in System Theory are:

•  Input synthesis: definition of an input function that generates, given an initial state, a desired state evolution or a desired output function

• Synthesis of the input and of initial state: determination of an input function and of an initial state that generate a desired state evolution or a desired output function

• Control synthesis: design and implementation of a device that, properly connected to the system, allows achieving desired properties and behaviors in terms of stability, disturbance rejection, output properties, …

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Systems and Models

• Concept of state for a dynamic system The “state” is the information on the “internal condition” of a dynamic system that is needed in each instant in order to predict the effect of the previous history of the system itself on its future behavior

•  In physical systems, the “internal condition” is typically determined by an

energy storage (also momentum or mass)

•  It is advisable, in the definition of the state variables, to choose variables related to these storages (e.g. the voltage of a capacitor or of an inductor in an electrical circuit, the velocity of a mass in a mechanical system, …)

• The state variables and the state equations are NOT defined uniquely!

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Systems and Models

Considerations related to storage/transmission of energy In each physical domain of interest, excluding the thermal one, there are two “components” that are able to store energy:

•  Electrical •  Capacitor (C) and Inductor (L)

•  Mechanical (linear) •  Mass (M) and compliance (1/K)

•  Mechanical (rotational) •  Inertia (J) and rotational compliance (1/K)

•  Fluid flow (hydraulic/pneumatic) •  Fluid Capacitor (Cf) and Fluid Inductor (Lf)

•  Thermal •  Thermal Capacitor (Ct)

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Considerations related to storage/transmission of energy The two basic elements to store energy:

Systems and Models

Effort variables are computed as “differences”

Domain “capacitive” storage “inductive” storage

E Cv=12

2 E Li=12

2electrical

mechanical (linear) E Mv=

12

2 EKf=

121 2

mechanical (rotational)

E J=12

2ω EKc=

121 2

hydraulic/pneumatic E C pf=12

2 E L qf=12

2

thermic E C Tt= Non present

Flow variables Effort variables The stored energy depends on the

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Systems and Models

Definition • The state of a dynamic system Σ:

•  Is an element of the state set X •  It may change in time,

•  The state x(t0) at the instant t0, together with the segment of the input function u|[t0,t1], uniquely defines the output function y|[t0,t1]

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Systems and Models - Examples

• RLC circuit

Choice of the state variables Quantities related to energy storage:

i vc

R

vu(t)

L

C vi(t) i

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R

vu(t)

L

C vi(t) i

2 states 1 output

• RLC circuit

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• Electric circuit with more loops

R1 R2

R3 C2

C1

vi(t) vu(t) v2

v1

i1 i2 i3

i5

i4

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R1 R2

R3 C2

C1

vi(t) vu(t) v2

v1

i1 i2 i3

i5

i4

2 states 1 output

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Output discontinuity due to the presence of the term D in the output equation

0 0.5 1 1.5 2 -0.8 -0.6 -0.4 -0.2

0 0.2 0.4 0.6 0.8

1 1.2 States x1 and x2

0 0.5 1 1.5 2 -0.8 -0.6 -0.4 -0.2

0 0.2 0.4 0.6 0.8

1 1.2 Input and output


R1 R2

R3 C2

C1

vi(t) vu(t) v2

v1

i1 i2 i3

i5

i4

The time-evolution of the state is a continuous function!

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• Mechanical system f(t) k2

m1

x1(t)

m2

x2(t) k1

b1

b2

4 states 2 output

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0 5 10 15 20 25 30 35 40 45 50 -1

-0.5

0

0.5

1

1.5

2

Tempo

Position of the two masses

x1

x2

f

m1 = 10; m2 = 5; k1 = 100; k2 = 50;

b1 = 10; b2 = 20;

0 10 20 30 40 50 -1

-0.5

0

0.5

1

1.5

2

Tempo

Position of the two masses

x1

x2

f

b1 = 50; b2 = 20;

f = 100 N

f(t) k2

m1

x1(t)

m2

x2(t) k1

b1

b2

• Mechanical system

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• DC electric motor

Ra La Cm(t), ω(t)

Le

va(t)

ia(t)

ve(t)

ie(t)

vc(t)

Cr(t)

u y

x

3 states 2 input 1 output

if ve = const

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• Sequential circuit (finite state system)

• Mathematical model

u1

sync

y u2

The system is synchronous: output is 1 if the current input symbol is 01 and if previously, between symbols 00 and 11, the last was 11.

1 1 1 0 1 1 0 0 0 0

11 10 01 00

0 0 1 0 1 0 0 0 0 0

11 10 01 00

u1 u2

u1 u2

x

x

State transition function

Output function

Discrete-time dynamic system

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• Another case of discrete time systems is obtained e.g. when a continuous time system is controlled by a digital computer: sampled data

u(k)

y(k) T

T

R1


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• Another case of discrete time systems is obtained e.g. when a continuous time system is controlled by a digital computer: sampled data

R1


u

t

u(0) u(1)

u(2)

u(3) u(4)

0 T 2T 3T 4T

•  If the input is a piecewise constant signal, and if the output is sampled at the same instants of time kT in which the input changes, then:

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• The state transition function in this case is given by

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Systems and Models - Definitions

•  In general, the mathematical model of a system is characterized by: 1.  The time set 2.  The set of input variables 3.  The set of input functions 4.  The set of output variables 5.  The set of state variables

•  D.1 A continuous-time system (continuous system) is obtained when

•  D.2 A discrete-time system (discrete system) is obtained when

Set of real numbers

Set of integer numbers

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Definition:

•  A model (system) is causal (nonanticipative or physical) when the output depends only on past and current values of the input

•  A non-causal model (system) is called anticipative or acausal.

•  An anticipative model does not correspond to any physical system •  It is not conceivable a system reacting to an input before it is applied.

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is not causal if x is the input and y is the output of the system. The derivative function is defined as ⇒ it is necessary to know the future value of the function!

It is not possible to implement an ideal derivative function

The model

The model is causal if y is the input and x the output

Non causal models are used sometimes for analysis and algebraic manipulation purposes.

y(t) = a d x(t)d t

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Causal systems and models

• Example (non physical system):

•  The amplitude (then the energy) of the output signal y(t) would increase (to infinite) if

the frequency ω of the input is increased!

0 2 4 6 8 10 -4 -2 0 2 4 A = 1, ω = 2 rad/sec

0 2 4 6 8 10 -4 -2 0 2 4 A = 1, ω = 4 rad/sec

Tempo (sec)

x(t) y(t)

limh!0

x(t+ h)! x(t)

h

d x(t)

d t

1

x(t) = A sinωt

y(t) =d x(t)

d t= A ω cosωt

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•  D.3 A system is called memory-less (or purely algebraic) if it is composed by the sets and by an input/output function

•  D.4 A system is called continuous-time dynamic system if it is composed by the sets , by a state differential equation

which admits a unique solution for each initial state and for each admissible input function, and by an output function

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•  D.5 A system is called discrete-time dynamic system if it is composed by the sets , by a future state equation

and by an output function •  D.6 A system is purely dynamic if its output function is

expressed as

i.e. it does not depend directly on the input

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u z

y

Purely dynamic system

Purely algebraic system

Non present in purely dynamic systems

n  Separation principle: dynamic part (with state) algebraic part

n  D.7 A system is called time-invariant or stationary if the time variable does not appear explicitly in the equations of its mathematical model. Otherwise, it is called non-stationary or time-varying.

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Time invariant models

•  If the properties of a system do not depend on the time (i.e. they are time invariant), then the corresponding parameters are constant. The models of these systems are called stationary or time-invariant.

•  In these cases, experiments are repeatable: the output obtained by applying a given input at time t0 with a given initial state x0 is the same (except a translation in time) obtained by applying, with the same initial state x0, the same input at time t0-δ.

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Tempo (s)

x, y

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Tempo (s)

x, y

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Time invariant models • From a practical point of view, usually the parameters of physical

systems change in time!

• On the other hand, it is sufficient that they do not change in a significant manner in a time period comparable with the `experiment’ duration.

•  In time-invariant models, the initial time t0 is not important, and therefore the value t0 = 0 is usually considered.

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•  D.8 A system is linear if 1.  The sets U, Uf , X, and Y are vector space (defined on the

same field T ) 2.  The functions defining its mathematical model are linear in x, u

for all the admissible t. Otherwise, a system is non-linear.

Commonly, a system is defined in mathematical term by:

Linear systems Non linear systems

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Purely algebraic system

Continuous-time dynamic system

Discrete-time dynamic system

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In order to take into account the physical limitation of the variables, the input set U is often considered limited

Examples:

•  Input voltage for an electric motor -Va < v < Va

•  Input flow in a tank 0 < u < U •  …

u1

u2

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Classification of the systems on basis of the state

D.9 It is then possible to define: •  Finite-State systems (state set: finite set)

•  Finite-Dimensional systems (state set: finite-dimensional vector space)

•  Infinite-Dimensional systems (state set: infinite-dimensional vector space)

= set of the states

Finite set

Vector space with Finite dimension

Infinite dimension

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State transition function

The (unique) solution of the equations has the form

and it is called the state transition function

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Properties of the state transition function 1)  Time orientation: it is defined for t ≥ t0, but not necessarily for t < t0

2)  Causality: its dependence on the input function is restricted to the time interval [t0, t]:

ϕ(t, t0, x0, u1(.)) = ϕ(t, t0, x0, u2(.)) if u1|[t0,t] = u2|[t0,t]

3)  Consistency: x = ϕ (t, t, x, u(.))

4)  Composition: consecutive state transitions are congruent. i.e.,

ϕ(t, t0, x0, u(.)) = ϕ(t, t1, x1, u(.)) provided that

x1 = ϕ(t1 , t0 , x0 , u(.)) t0 ≤ t ≤ t1

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The last property (composition) means that

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Representation of the time evolution of a system •  The time evolution of the state can be represented as a

trajectory in the state space, parameterized as a function of time.

x2 x1

x3

x(0)

t t1

t2

t3

t

u

t1 t2 t3

Trajectory: {x(t) : x(t) = ϕ(t , t0 , x(t0 ), u(.)), t ∈ [t0 , t1 ]}

Input function The choice of the input in a given instant of time allows to define different orientations to the tangential direction of the trajectory at t: the input affects the velocity!

Possible control choices

x(t) = ϕ(t , t0 , x(t0 ), u(.))

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•  Event: pair {t, x(t)} ∈ T x X •  Motion or Movement for t ∈ [t0, t1] is the set of events defined by the

transition function

The motion is defined in the space T x X

•  Trajectory: image in X of the transition function for t ∈ [t0, t1]

The trajectory is defined in X

0 2 4 6 8 10

0 1

2 3 0

0.5 1

1.5 2

t x1"

x 2"

Trajectory

Motion

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•  For any given initial state, different input functions cause different trajectories, all initiating at the same point of the state space;

•  Selecting input at a particular instant of time (for instance, t3) allows different orientations in space of the tangent to the trajectory at t3, namely of the state velocity

x2 x1

x3

x(0)

t t1

t2

t3

t

u

t1 t2 t3

x1, x2, x3

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•  In a similar way, it is possible to obtain a function describing the output trajectories. From:

One obtains:

From (1) we have obtained the state transition function that, substituted in (2), gives the response function

Output trajectory:

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In conclusion: •  A system, once the differential equation of its

mathematical model has been solved, is characterized by a state transition function

•  and by a response function

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•  D.10 Indistinguishable states in [t0, t1] : the states x1 , x2 ∈ X are indistinguishable if

•  D.11 Equivalent states: the states x1 , x2 ∈ X are equivalent if they are indistinguishable for each pair of instants

•  D.12 Minimal system: is a dynamic system without equivalent states

A non-minimal system can be put in minimal form by defining a new state set in which each state corresponds to a class of equivalent old states.

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•  Example:

vu(t)

vi(t)

R R

R R

R1

C

vu(t)

vi(t) R

R1

Non-minimal system Minimal system

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•  Example:

θ(t), ω(t) va(t)

cr(t)

q(t)

z

Motor

Pump

Pump: cr = kp ω, q = kq ω Tank: dz / dt = ks q

When the two models are considered together, two equivalent states are identified (θ, z). One of them can be ignored: in this case θ since we want to control the liquid level only.

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•  D.13 Equivalent systems: the systems Σ1 and Σ2 are equivalent if they are compatible, i.e.

and if to any state x1 ∈ X (x2 ∈ X) of Σ1 it is possible to associate a state x2 ∈ X (x1 ∈ X) of Σ2 (and vice versa) such that

Σ1

Σ2

u(.)

x1

x2

y1

y2

limh!0

x(t+ h)" x(t)

h

d x(t)

d t

!(t, t0, x0, u1(·)) = !(t, t0, x0, u2(·)) if u1|[t0,t] = u2|[t0,t]

T1 = T2 = TU1 = U2 = UUf1 = Uf2 = Uf

Y1 = Y2 = Y

1

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•  Example:

•  The two systems are equivalent if

•  The initial states must be such that

R1 vu(t)

L

vi(t) i0

R2

C

vi(t) vu(t) vc

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•  As a matter of fact:

R1 vu(t)

L

vi(t) i0

R2

C vi(t) vu(t) vc

The two systems are equivalent if

The evolution of the states is such that

1

limh!0

x(t+ h)! x(t)

h

d x(t)

d t


T1 = T2 = TU1 = U2 = UUf1 = Uf2 = Uf

Y1 = Y2 = Y

!

x1 = !R1

Lx1 + 1

Lu

y = R1x1

!

x2 = ! 1CR2

x2 + 1CR2

u

y = x2

1

limh!0

x(t+ h)! x(t)

h

d x(t)

d t


T1 = T2 = TU1 = U2 = UUf1 = Uf2 = Uf

Y1 = Y2 = Y

!

x1 = !R1

Lx1 + 1

Lu

y = R1x1

!

x2 = ! 1CR2

x2 + 1CR2

u

y = x2

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•  D.14 Temporary equilibrium state. In a dynamic system Σ , a state is a temporary equilibrium state in [t0, t1] if there exists an input function

such that

Among all the possible motions, particular and important motions are those constant. The corresponding trajectory results in a unique state, called equilibrium state.

f

m

s

s0

input

M = M0 + m t Mass 0 ≤ f ≤ F Force

Notice: equilibrium depends on the input!!

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•  D.15 Equilibrium state: it is a temporary equilibrium state in [t0, t1] for any pair t0, t1 in T

•  Example:

f s

s0

input

M = M0 Mass 0 ≤ f ≤ F Force

N.B. It may happen that not all the states s can be equilibrium states, because f is limited

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•  Mathematical models:

•  Particular cases: stationary models, linear models, linear stationary models

Input-state-output Differential model

Input-state-output Difference model

Input-state-output Global model

External Global model

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Time invariant systems (time-shifting of causes and effects) Time invariant systems satisfy the time-shifting of causes and effects property.

In time-invariant systems, the transition function and the response function verify the following equations:

t τ 0

u(t) uΔ(t)

uΔ(t) = u(t-τ) Shifted input function

Assume that

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In particular, if τ = -t0, one otains

Therefore, for time invariant systems: 1.  It is always possible to consider as initial istant of time t0 = 0 2.  The transition and the response functions depend in a linear

fashion on the difference t - t0 , not on t and t0 separately

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Linear systems

•  Functions ϕ and γ are linear with respect to initial state and input function

Let consider two scalars α and β, and

In the particular case α = β = 1, these equations correspond to the so-called property of superposition of the effects.

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•  Important consequence: for linear systems, the property of decomposability of state transition and response functions holds.

•  Moreover: 1.  Indistinguishable states in [t0, t1] generate the same free

response in [t0, t1] 2.  A Linear System is in minimal form iff for any initial instant of

time t0 no different states generate the same free response

Free motion Forced motion

Free response Forced response

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Controllability and Reachability The term controllability denotes the possibility of influencing the motion x(.) or the response y(.) of a dynamical system Σ by means of the input function (or control function) u(.)∈Uf . It might be required to steer a system from a state x0 to x1 or from an event (t0, x0) to (t1, x1): if this is possible, the system is said to be controllable from x0 to x1 or from (t0, x0) to (t1, x1). Equivalent statements: •  “the state x0 (or the event (t0, x0)) is controllable to x1 (or to (t1, x1))” and •  “the state x1 (or the event (t1, x1)) is reachable from x0 (or from (t0, x0))”

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Controllability and Reachability Controllability analysis is strictly connected to the definition of particular subsets of the state space X, that is: D.16 Set of the states reachable at time instant t1 from the state x0 at t0 (i.e. from the event (t0 , x0) ) D.17 Set of the states reachable at any time belonging to the interval [t0, t1] from the state x0 at t0 (i.e. from the event (t0 , x0) )

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D.18 Set of the states controllable at the event (t1 , x1) from the initial time t0

D.19 Set of the states controllable at the event (t1 , x1) from any time in [t0, t1]

A property:

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Representation in X = R2 Set of the events (t, x)

t t0 t1

x(t1)

Events (t,x) corresponding to time t0, t1

x(t0)

Trajectory x(t) corresponding to [t0 t1]

x’(t0)

x”(t0) x’(t1)

x”(t1)

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t

These are the set of all the admissible motions with (t1, x1) or (t0, x0) as final or initial event.

t0 t1

x0

x1

Representation in X = R2 Set of the events (t, x)

Events (t,x) corresponding to time t0, t1

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Let’s define:

Then

Moreover:

Projection along t on P0 of

Projection along t on P1 of

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D.20 The state set of a dynamic system Σ or, by extension, system Σ itself, is said to be completely reachable from the event (t0, x) in the

time interval [t0, t1] if

D.21 The state set of a dynamic system Σ or, by extension, system Σ itself, is said to be completely controllable to the event (t1, x) in the time interval [t0, t1] if

t t0 t1 x

t t0 t1

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In case of time invariant systems, since we can assume t0 = 0, a simplified notation can be used :

The sets and are defined as These are the state sets reachable from z and controllable to z in an arbitrarily long period of time.

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• D.22 A time-invariant system is completely controllable (or connected) if it is possible to reach any state from any other state, so that W+(x)=W−(x)=X for all x∈X.

t t0 t1 x0

x1 x0

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Observability (and Reconstructability) The term observability denotes the possibility of deriving the initial state x(t0) or the final state x(t1) of a dynamic system Σ when the time evolutions of input and output in the time interval [t0, t1] are known. Final state observability is denoted also with the term reconstructability. The state observation and reconstruction problems may not always admit a solution: this happens, in particular, for observation when the initial state belongs to a class whose elements are indistinguishable in[t0, t1].

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Observability (and Reconstructability) Let’s define: • The set of all the initial states consistent with the functions u(.), y(.) in

the time interval [t0, t1]

• The set of all the final states consistent with the functions u(.), y(.) in the time interval [t0, t1]

where y(.) is not arbitrary, but constrained to the set of all the output functions admissible with respect to the initial state and the input function. This set is defined by

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The state set of a dynamic system Σ or, by extension, system Σ itself, is • Observable in [t0, t1] by a suitable experiment (called

diagnosis) if there exists at least one input function u(.)∈Uf such that reduces to a single element for all y(.) ∈ Yf (t0, u(.));

• Reconstructable in [t0, t1] by a suitable experiment (called homing) if there exists at least one input function u(.)∈ Uf such that reduces to a single element for all y(.)∈Yf (t0, u(.)).

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The state set of a dynamic system Σ or, by extension, system Σ itself, is • Fully observable in [t0, t1] if it is observable in [t0, t1] with any

input function u(.)∈Uf;

• Fully reconstructable in [t0, t1] ] if it is reconstructable in [t0, t1] with any input function u(.)∈Uf;

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•  With time invariant system, we can adopt the notation

•  Then:

Observable system

Reconstructable system

Fully reconstructable

system

Fully observable

system

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Systems and Models

The above sets are often considered in solving problems related to system control and observation. The most significant of these problems are:

1.  Control between two given states: given two states x0 and x1 at two instants of time t0 and t1, where t1 > t0, determine an input function u(.) such that x1 =ϕ(t1, t0, x0, u(.)).

2.  Control to a given output: given an initial state x0, an output value y1 at two instants of time t0, t1, where t1 > t0, determine an input u(.) such that y1 =γ(t1, t0, x0, u(.)).

3.  Control for a given output function: given an initial state x0, an admissible output function y(.) at two instants of time t0, t1, where t1 > t0, determine an input u(.) such that y(t)=γ(t, t0, x0, u(.)) for all t∈[t0, t1].

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Systems and Models

4.  State observation: given corresponding input and output functions u(.), y(-) at two instants of t0, t1, where t1 > t0, determine an initial state x0 (or the whole set of initial states) consistent with them, i.e., such that y(t)=γ(t, t0, x0, u(.)) for all t∈[t0, t1].

5.  State reconstruction: given corresponding input and output functions u(.), y(.) at two instants of time t0, t1, where t1 > t0, determine a final state x1 (or the whole set of final states) consistent with them, i.e., corresponding to an initial state x0 such that x1 =ϕ(t1, t0, x0, u(.)), y(t)=γ(t, t0, x0, u(.)) for all t∈[t0, t1].

6.  Diagnosis: like 4, except solution also includes the choice of a suitable input function.

7.  Homing: like 5, except solution also includes the choice of a suitable input function.

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AUTOMATIC CONTROL AND SYSTEM THEORY - … · AUTOMATIC CONTROL AND SYSTEM THEORY ... • Physical properties are usually distributed in the systems: ... approximations that allow