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: claudio.melchiorri@unibo.it
C. Melchiorri (DEI) Automatic Control & System Theory 2 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
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 4: connections 1 3 4 External environment C. Melchiorri (DEI) Automatic Control & System Theory 3
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 C. Melchiorri (DEI) Automatic Control & System Theory 4
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. C. Melchiorri (DEI) Automatic Control & System Theory 5
C. Melchiorri (DEI) Automatic Control & System Theory 6 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 S 1 S 2
C. Melchiorri (DEI) Automatic Control & System Theory 7 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 R i a L a (t) a c(t), ω(t) i e (t) input u 1 (t) u 2 (t) S u 3 (t) y(t) output v a (t) v e (t) L e
C. Melchiorri (DEI) Automatic Control & System Theory 8 Systems and Models Systems Distributed Parameters Lumped Parameters Stochastic Deterministic Continuous time Discrete time Nonlinear Linear Time Varying Constant coefficients Non homogeneous Homogeneous
C. Melchiorri (DEI) Automatic Control & System Theory 9 Systems and Models Systems Distributed Parameters Lumped Parameters Stochastic Deterministic Continuous time Discrete time Nonlinear Linear Time Varying Constant coefficients Non homogeneous Homogeneous
C. Melchiorri (DEI) Automatic Control & System Theory 10 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
C. Melchiorri (DEI) Automatic Control & System Theory 11 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. Concentrated mass Concentrated elasticity 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.
C. Melchiorri (DEI) Automatic Control & System Theory 12 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:
C. Melchiorri (DEI) Automatic Control & System Theory 13 Systems and Models 60 Static system (algebraic) i(t) V, I 50 40 30 20 v(t) R v(t) =Ri(t) 10 0 0 50 100 150 200 250 300 Time (s) Dynamic system 1 0.8 0.6 0.4 q i(t) C 0.2 v 0 0 0.5 1 1.5 2 v(t) R
C. Melchiorri (DEI) Automatic Control & System Theory 14 Systems and Models Static systems Electric circuit with resistances R 1 v i (t) R 2 v u (t) Combinational circuit u 1 u 2 u 3 y u 1 u 2 u 3 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 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. 1 1 1 1
C. Melchiorri (DEI) Automatic Control & System Theory 15 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) R 1 u(t) R 2 v u (t) v c (t) x u(t) input v u (t) output x state
C. Melchiorri (DEI) Automatic Control & System Theory 16 Systems and Models Dynamic systems Electrical circuit with elements able to store energy (capacitors and/or inductors) R 1 u(t) R 2 v u (t) v c (t) x u(t) input v u (t) output One obtains x state
C. Melchiorri (DEI) Automatic Control & System Theory 17 Systems and Models The description of the system is made with two equations, a differential and an algebraic one: State differential equation Output equation u x y
C. Melchiorri (DEI) Automatic Control & System Theory 18 Systems and Models The description of the system is made with two equations, a differential and an algebraic one: State differential equation Output equation 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) = x 0 (value of the state at t = 0), one obtains: State transition equation Similar equations hold also in the discrete time case.
C. Melchiorri (DEI) Automatic Control & System Theory 19 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
C. Melchiorri (DEI) Automatic Control & System Theory 20 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) = x 0 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
C. Melchiorri (DEI) Automatic Control & System Theory 21 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,
C. Melchiorri (DEI) Automatic Control & System Theory 22 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!
C. Melchiorri (DEI) Automatic Control & System Theory 23 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 (C f ) and Fluid Inductor (L f ) Thermal Thermal Capacitor (C t )
Systems and Models Considerations related to storage/transmission of energy The two basic elements to store energy: Domain capacitive storage inductive storage electrical E = 1 Cv 2 2 E = 1 2 Li 2 mechanical (linear) E = 1 2 Mv 2 E = 1 2 1 2 K f mechanical (rotational) E = 1 2 Jω 2 E = 1 2 1 2 K c hydraulic/pneumatic E = 1 2 C f p 2 E = 1 2 L q f 2 thermic E = C t T Non present The stored energy depends on the Effort variables Flow variables Effort variables are computed as differences C. Melchiorri (DEI) Automatic Control & System Theory 24
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(t 0 ) at the instant t 0, together with the segment of the input function u [t0,t1], uniquely defines the output function y [t0,t1] C. Melchiorri (DEI) Automatic Control & System Theory 25
C. Melchiorri (DEI) Automatic Control & System Theory 26 Systems and Models - Examples RLC circuit L R Choice of the state variables v i (t) i C v u (t) Quantities related to energy storage: i v c
C. Melchiorri (DEI) Automatic Control & System Theory 27 Systems and Models - Examples RLC circuit L R v i (t) i C v u (t) 2 states 1 output
C. Melchiorri (DEI) Automatic Control & System Theory 28 Systems and Models - Examples Electric circuit with more loops C 1 R 1 R 2 v 1 i 4 v i (t) i 1 i 2 i 3 i 5 v 2 C 2 R 3 v u (t)
Systems and Models - Examples Electric circuit with more loops C 1 R 1 R 2 v 1 i 4 v i (t) i 1 i 2 i 3 i 5 v 2 C 2 R 3 v u (t) 2 states 1 output C. Melchiorri (DEI) Automatic Control & System Theory 29
C. Melchiorri (DEI) Automatic Control & System Theory 30 Systems and Models - Examples Electric circuit with more loops C 1 R 1 R 2 v 1 i 4 v i (t) i 1 i 2 i 3 i 5 v 2 C 2 R 3 v u (t) The time-evolution of the state is a continuous function! Output discontinuity due to the presence of the term D in the output equation 1.2 Input and output 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 0 0.5 1 1.5 2 1.2 States x1 and x2 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 0 0.5 1 1.5 2
C. Melchiorri (DEI) Automatic Control & System Theory 31 Systems and Models - Examples Mechanical system f(t) k 2 b 1 k 1 m 1 x 1 (t) m 2 x 2 (t) b 2 4 states 2 output
Systems and Models - Examples Mechanical system m1 = 10; m2 = 5; k1 = 100; k2 = 50; 2 b1 = 10; b2 = 20; f(t) k 2 b 1 m 1 m 2 k 1 b 2 x 1 (t) x 2 (t) Position of the two masses Position of the two masses 2 f = 100 N b1 = 50; b2 = 20; 1.5 f 1.5 f 1 x 1 1 x 1 0.5 0.5 0 0 x 2 x 2-0.5-0.5-1 0 5 10 15 20 25 30 35 40 45 50 Tempo -1 0 10 20 30 40 50 Tempo C. Melchiorri (DEI) Automatic Control & System Theory 32
Systems and Models - Examples DC electric motor i a (t) R a L a C m (t), ω(t) v a (t) i e (t) v e (t) L e v c (t) C r (t) if v e = const u y x 3 states 2 input 1 output C. Melchiorri (DEI) Automatic Control & System Theory 33
C. Melchiorri (DEI) Automatic Control & System Theory 34 Systems and Models - Examples Sequential circuit (finite state system) u 1 u 2 sync Mathematical model y 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. x u 1 u 2 0 1 00 0 0 01 0 1 10 0 1 11 1 1 State transition function u 1 u 2 x 0 00 0 01 0 10 0 11 0 Output function Discrete-time dynamic system 1 0 1 0 0
C. Melchiorri (DEI) Automatic Control & System Theory 35 Systems and Models - Examples Another case of discrete time systems is obtained e.g. when a continuous time system is controlled by a digital computer: sampled data R 1 u(t) R 2 v u (t) v c (t) x T u(k) y(k) T
C. Melchiorri (DEI) Automatic Control & System Theory 36 Systems and Models - Examples Another case of discrete time systems is obtained e.g. when a continuous time system is controlled by a digital computer: sampled data R 1 u(t) R 2 v u (t) v c (t) x 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: u u(2) u(4) u(1) u(3) u(0) 0 T 2T 3T 4T t
C. Melchiorri (DEI) Automatic Control & System Theory 37 Systems and Models - Examples The state transition function in this case is given by
C. Melchiorri (DEI) Automatic Control & System Theory 38 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 Set of real numbers D.2 A discrete-time system (discrete system) is obtained when Set of integer numbers
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. C. Melchiorri (DEI) Automatic Control & System Theory 39
C. Melchiorri (DEI) Automatic Control & System Theory 40 The model y(t) =a dx(t) dt 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 is causal if y is the input and x the output Non causal models are used sometimes for analysis and algebraic manipulation purposes.
Causal systems and models Example (non physical system): x(t) = A sin ωt y(t) = dx(t) = A ω cos ωt dt x(t) dx(t) dt y(t) The amplitude (then the energy) of the output signal y(t) would increase (to infinite) if the frequency ω of the input is increased! 4 2 A = 1, ω = 2 rad/sec 0-2 -4 0 2 4 6 8 10 4 A = 1, ω = 4 rad/sec 2 0-2 -4 0 2 4 6 8 10 Tempo (sec) C. Melchiorri (DEI) Automatic Control & System Theory 41
C. Melchiorri (DEI) Automatic Control & System Theory 42 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
C. Melchiorri (DEI) Automatic Control & System Theory 43 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
C. Melchiorri (DEI) Automatic Control & System Theory 44 u Purely dynamic system z Purely algebraic system y 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 timevarying.
C. Melchiorri (DEI) Automatic Control & System Theory 45 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 t 0 with a given initial state x 0 is the same (except a translation in time) obtained by applying, with the same initial state x 0, the same input at time t 0 -δ. 2.5 2.5 2 2 1.5 1.5 1 1 x, y 0.5 x, y 0.5 0 0-0.5-0.5-1 -1-1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Tempo (s) -1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Tempo (s)
C. Melchiorri (DEI) Automatic Control & System Theory 46 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 t 0 is not important, and therefore the value t 0 = 0 is usually considered.
D.8 A system is linear if 1. The sets U, U f, 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 C. Melchiorri (DEI) Automatic Control & System Theory 47
C. Melchiorri (DEI) Automatic Control & System Theory 48 Purely algebraic system Continuous-time dynamic system Discrete-time dynamic system
In order to take into account the physical limitation of the variables, the input set U is often considered limited u 2 u 1 Examples: Input voltage for an electric motor -V a < v < V a Input flow in a tank 0 < u < U C. Melchiorri (DEI) Automatic Control & System Theory 49
C. Melchiorri (DEI) Automatic Control & System Theory 50 Classification of the systems on basis of the state Finite set = set of the states Vector space with Finite dimension Infinite dimension 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)
C. Melchiorri (DEI) Automatic Control & System Theory 51 State transition function The (unique) solution of the equations has the form and it is called the state transition function
Properties of the state transition function 1) Time orientation: it is defined for t t 0, but not necessarily for t < t 0 2) Causality: its dependence on the input function is restricted to the time interval [t 0, t]: ϕ(t, t 0, x 0, u 1 (.)) = ϕ(t, t 0, x 0, u 2 (.)) if u 1 [t0,t] = u 2 [t0,t] 3) Consistency: x = ϕ (t, t, x, u(.)) 4) Composition: consecutive state transitions are congruent. i.e., ϕ(t, t 0, x 0, u(.)) = ϕ(t, t 1, x 1, u(.)) provided that x 1 = ϕ(t 1, t 0, x 0, u(.)) t 0 t t 1 C. Melchiorri (DEI) Automatic Control & System Theory 52
C. Melchiorri (DEI) Automatic Control & System Theory 53 The last property (composition) means that
C. Melchiorri (DEI) Automatic Control & System Theory 54 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. x(t) = ϕ(t, t 0, x(t 0 ), u(.)) t x 3 t 1 t 2 t 3 Trajectory: x(0) x1 x 2 {x(t) : x(t) = ϕ(t, t 0, x(t 0 ), u(.)), t [t 0, t 1 ]} u Possible control choices t 1 t 2 t 3 t 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!
Event: pair {t, x(t)} T x X Motion or Movement for t [t 0, t 1 ] 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 [t 0, t 1 ] The trajectory is defined in X 2 x 2 " 1.5 1 Motion x 1 " t C. Melchiorri (DEI) Automatic Control & System Theory 55 0.5 0 3 Trajectory 2 1 0 0 2 4 6 8 10
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, t 3 ) allows different orientations in space of the tangent to the trajectory at t 3, namely of the state velocity ẋ 1, ẋ 2, ẋ 3 t x 3 t 1 t 2 t 3 x(0) x1 x 2 u t 1 t 2 t 3 C. Melchiorri (DEI) Automatic Control & System Theory 56 t
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: C. Melchiorri (DEI) Automatic Control & System Theory 57
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 C. Melchiorri (DEI) Automatic Control & System Theory 58
C. Melchiorri (DEI) Automatic Control & System Theory 59 D.10 Indistinguishable states in [t 0, t 1 ] : the states x 1, x 2 X are indistinguishable if D.11 Equivalent states: the states x 1, x 2 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.
C. Melchiorri (DEI) Automatic Control & System Theory 60 Example: R C R R v i (t) R R v i (t) R 1 v u (t) R 1 v u (t) Non-minimal system Minimal system
C. Melchiorri (DEI) Automatic Control & System Theory 61 Example: Motor Pump v a (t) θ(t), ω(t) c r (t) q(t) z Pump: c r = k p ω, q = k q ω Tank: dz / dt = k s 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.
D.13 Equivalent systems: the systems Σ 1 and Σ 2 are equivalent if they are compatible, i.e. T 1 = T 2 = T U 1 = U 2 = U U f1 = U f2 = U f Y 1 = Y 2 = Y and if to any state x 1 X (x 2 X) of Σ 1 it is possible to associate a state x 2 X (x 1 X) of Σ 2 (and vice versa) such that u(.) Σ 1 x 1 y 1 y 2 Σ 2 x 2 C. Melchiorri (DEI) Automatic Control & System Theory 62
C. Melchiorri (DEI) Automatic Control & System Theory 63 Example: L R 2 v i (t) i 0 R 1 v u (t) v i (t) v c C v u (t) The two systems are equivalent if The initial states must be such that
C. Melchiorri (DEI) Automatic Control & System Theory 64 As a matter of fact: L R 2 v i (t) i 0 R 1 v u (t) v i (t) v c C v u (t) {{ ẋ1 = R 1 L x 1 + 1 L u y = R 1 x 1 { ẋ2 = 1 CR 2 x 2 + 1 CR 2 u y = x 2 The two systems are equivalent if The evolution of the states is such that
C. Melchiorri (DEI) Automatic Control & System Theory 65 Among all the possible motions, particular and important motions are those constant. The corresponding trajectory results in a unique state, called equilibrium state. D.14 Temporary equilibrium state. In a dynamic system Σ, a state is a temporary equilibrium state in [t 0, t 1 ] if there exists an input function such that m M = M 0 + m t Mass s f input 0 f F Force s 0 Notice: equilibrium depends on the input!!
C. Melchiorri (DEI) Automatic Control & System Theory 66 D.15 Equilibrium state: it is a temporary equilibrium state in [t 0, t 1 ] for any pair t 0, t 1 in T Example: s f input M = M 0 Mass 0 f F Force s 0 N.B. It may happen that not all the states s can be equilibrium states, because f is limited
C. Melchiorri (DEI) Automatic Control & System Theory 67 Mathematical models: Input-state-output Differential model Input-state-output Difference model Input-state-output Global model External Global model Particular cases: stationary models, linear models, linear stationary models
C. Melchiorri (DEI) Automatic Control & System Theory 68 Time invariant systems (time-shifting of causes and effects) Time invariant systems satisfy the time-shifting of causes and effects property. u(t) u Δ (t) u Δ (t) = u(t-τ) Assume that Shifted input function 0 τ t In time-invariant systems, the transition function and the response function verify the following equations:
In particular, if τ = -t 0, one otains Therefore, for time invariant systems: 1. It is always possible to consider as initial istant of time t 0 = 0 2. The transition and the response functions depend in a linear fashion on the difference t - t 0, not on t and t 0 separately C. Melchiorri (DEI) Automatic Control & System Theory 69
C. Melchiorri (DEI) Automatic Control & System Theory 70 Linear systems Functions ϕ and 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 socalled property of superposition of the effects.
C. Melchiorri (DEI) Automatic Control & System Theory 71 Important consequence: for linear systems, the property of decomposability of state transition and response functions holds. Free motion Forced motion Free response Forced response Moreover: 1. Indistinguishable states in [t 0, t 1 ] generate the same free response in [t 0, t 1 ] 2. A Linear System is in minimal form iff for any initial instant of time t 0 no different states generate the same free response
C. Melchiorri (DEI) Automatic Control & System Theory 72 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(.) U f. It might be required to steer a system from a state x 0 to x 1 or from an event (t 0, x 0 ) to (t 1, x 1 ): if this is possible, the system is said to be controllable from x 0 to x 1 or from (t 0, x 0 ) to (t 1, x 1 ). Equivalent statements: the state x 0 (or the event (t 0, x 0 )) is controllable to x 1 (or to (t 1, x 1 )) and the state x 1 (or the event (t 1, x 1 )) is reachable from x 0 (or from (t 0, x 0 ))
C. Melchiorri (DEI) Automatic Control & System Theory 73 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 t 1 from the state x 0 at t 0 (i.e. from the event (t 0, x 0 ) ) D.17 Set of the states reachable at any time belonging to the interval [t 0, t 1 ] from the state x 0 at t 0 (i.e. from the event (t 0, x 0 ) )
C. Melchiorri (DEI) Automatic Control & System Theory 74 D.18 Set of the states controllable at the event (t 1, x 1 ) from the initial time t 0 D.19 Set of the states controllable at the event (t 1, x 1 ) from any time in [t 0, t 1 ] A property:
C. Melchiorri (DEI) Automatic Control & System Theory 75 Representation in X = R 2 Set of the events (t, x) x (t 0 ) x(t 1 ) x(t 0 ) x (t 0 ) x (t 1 ) Trajectory x(t) corresponding to [t 0 t 1 ] x (t 1 ) t 0 t 1 t Events (t,x) corresponding to time t 0, t 1
Representation in X = R 2 Set of the events (t, x) x 1 x 0 t 0 t 1 t Events (t,x) corresponding to time t 0, t 1 These are the set of all the admissible motions with (t 1, x 1 ) or (t 0, x 0 ) as final or initial event. C. Melchiorri (DEI) Automatic Control & System Theory 76
C. Melchiorri (DEI) Automatic Control & System Theory 77 Let s define: Then Moreover: Projection along t on P 0 of Projection along t on P 1 of
C. Melchiorri (DEI) Automatic Control & System Theory 78 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 (t 0, x) in the time interval [t 0, t 1 ] if x t 0 t 1 t 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 (t 1, x) in the time interval [t 0, t 1 ] if t 0 t 1 t
C. Melchiorri (DEI) Automatic Control & System Theory 79 In case of time invariant systems, since we can assume t 0 = 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.
C. Melchiorri (DEI) Automatic Control & System Theory 80 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. x 0 x 1 x 0 t 0 t 1 t
C. Melchiorri (DEI) Automatic Control & System Theory 81 Observability (and Reconstructability) The term observability denotes the possibility of deriving the initial state x(t 0 ) or the final state x(t 1 ) of a dynamic system Σ when the time evolutions of input and output in the time interval [t 0, t 1 ] 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 [t 0, t 1 ].
Observability (and Reconstructability) Let s define: The set of all the initial states consistent with the functions u(.), y(.) in the time interval [t 0, t 1 ] The set of all the final states consistent with the functions u(.), y(.) in the time interval [t 0, t 1 ] 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 C. Melchiorri (DEI) Automatic Control & System Theory 82
C. Melchiorri (DEI) Automatic Control & System Theory 83 The state set of a dynamic system Σ or, by extension, system Σ itself, is Observable in [t 0, t 1 ] by a suitable experiment (called diagnosis) if there exists at least one input function u(.) U f such that reduces to a single element for all y(.) Y f (t 0, u(.)); Reconstructable in [t 0, t 1 ] by a suitable experiment (called homing) if there exists at least one input function u(.) U f such that reduces to a single element for all y(.) Y f (t 0, u(.)).
C. Melchiorri (DEI) Automatic Control & System Theory 84 The state set of a dynamic system Σ or, by extension, system Σ itself, is Fully observable in [t 0, t 1 ] if it is observable in [t 0, t 1 ] with any input function u(.) U f ; Fully reconstructable in [t 0, t 1 ] ] if it is reconstructable in [t 0, t 1 ] with any input function u(.) U f ;
C. Melchiorri (DEI) Automatic Control & System Theory 85 With time invariant system, we can adopt the notation Then: Observable system Reconstructable system Fully observable system Fully reconstructable system
C. Melchiorri (DEI) Automatic Control & System Theory 86 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 x 0 and x 1 at two instants of time t 0 and t 1, where t 1 > t 0, determine an input function u(.) such that x 1 =ϕ(t 1, t 0, x 0, u(.)). 2. Control to a given output: given an initial state x 0, an output value y 1 at two instants of time t 0, t 1, where t 1 > t 0, determine an input u(.) such that y 1 =γ(t 1, t 0, x 0, u(.)). 3. Control for a given output function: given an initial state x 0, an admissible output function y(.) at two instants of time t 0, t 1, where t 1 > t 0, determine an input u(.) such that y(t)=γ(t, t 0, x 0, u(.)) for all t [t 0, t 1 ].
C. Melchiorri (DEI) Automatic Control & System Theory 87 Systems and Models 4. State observation: given corresponding input and output functions u(.), y(-) at two instants of t 0, t 1, where t 1 > t 0, determine an initial state x 0 (or the whole set of initial states) consistent with them, i.e., such that y(t)=γ(t, t 0, x 0, u(.)) for all t [t0, t1]. 5. State reconstruction: given corresponding input and output functions u(.), y(.) at two instants of time t 0, t 1, where t 1 > t 0, determine a final state x 1 (or the whole set of final states) consistent with them, i.e., corresponding to an initial state x 0 such that x 1 =ϕ(t1, t0, x0, u(.)), y(t)=γ(t, t0, x0, u(.)) for all t [t 0, t 1 ]. 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.