Dynamical systems: basic concepts

Transcription

1 Dynamical systems: basic concepts Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Fondamenti di Automatica e Controlli Automatici A.A / 14

2 Mystery box: basic definitions Consider the following scheme u(t) P x(t) y(t) where u R p is the input (exogenous) signal, y R q is the output signal and x R n is the plant P state, for some integers p, q and n. We can say that through the plant P the signal y and the state x are in some (dynamical) relation with u. If t R 0 the plant, or system, P is defined in continuous time, then u( ) : R R p, x( ) : R R n, y( ) : R R q. If t Z the plant, or system, P is defined in discrete time, then u( ) : Z R p, x( ) : Z R n, y( ) : Z R q. There are possible modifications, as an example the input or the output, or even the state, could be constrained to assume integers value: signals are quantized. 2 / 14

3 Mystery box: the state of the system The state x(t) of the system is defined to be the set of variables that are necessary to completely the define the system output y(t) given an arbitrary input u(t). {x(t 0 ), u(τ) τ t 0 } P {}}{ = {y(t) t t 0 } The system P establishes a functional relation between u(t), x(t) and y(t). The information contained in the state x at certain time t 0 is sufficient to describe unequivocally the system evolution (y(t) and x(t)) for all t t 0 knowing the input signal u(t) for all t t 0. The past input history up to time t 0 is embedded in the state x(t 0 ). 3 / 14

4 An example: static plant Consider a plant consisting on a simple mass M that can slide along a straight horizontal frictionless surface and is subject to a force F (t) applied at its center of mass z c(t). M z c (t) F (t) z(t) By Newton s law we can write that M z c(t) = F (t). Pick z c(t) as the system output y(t) = z c(t) and define u(t) = F (t) as the system input, then y(t) = u(t) M. In this case the functional relation between y(t) and u(t) is instantaneous and there is no need to define a state for this system. The functional relation between u(t) and y(t), i.e. P, is said to be STATIC. 4 / 14

5 An example: dynamic plant Consider the previous plant, and assume that we measure y(t) = ż c(t) with u(t) = F (t), then z c(t) = ẏ(t) = u(t) M, (1) that is, the dependence of the output y(t) from the input u(t) is expressed via a differential equation. Then t t u(τ) ẏ(τ) dτ = y(t) y(t 0 ) = dτ (2) t 0 t 0 M t u(τ) y(t) = y(t 0 ) + dτ. (3) t 0 M Note that in this case to obtain y(t) the knowledge of the past input values since time t 0 are necessary (integral evaluation). The system is said to have MEMORY. Note also that it is necessary to know also the initial condition y(t 0 ). 5 / 14

6 An example: dynamic plant cont d Consider the previous plant, and assume that we measure y(t) = z c(t) with u(t) = F (t), then z c(t) = ÿ(t) = u(t) M, (4) that is, the dependence of the output y(t) from the input u(t) is expressed via a second order differential equation. Then t t u(τ) ÿ(τ) dτ = ẏ(t) ẏ(t 0 ) = dτ (5) t 0 t 0 M t t ẏ(τ) dτ tẏ(t 0 ) = t 0 t 0 u(τ) M t y(t) = y(t 0 ) + tẏ(t 0 ) + dτ (6) t 0 u(τ) M dτ. (7) In this case to obtain y(t) the knowledge of the past input values since time t 0, y(t 0 ) and ẏ(t 0 ) are necessary: also this system is said to have MEMORY. 6 / 14

7 An example: the state comes into play Define the state of the system as x 1 (t) = y(t), x 2 (t) = ẏ(t), x = [ ] x1, x 2 then the second order differential equation can be rewritten using a state x of dimension two by mean of a first order differential matrix equation such as ] [ ] [ ] [ ] [ẋ1 x ẋ = = u = x + 1 u = Ax + Bu, ẋ 2 M 0 0 M }{{}}{{} A B with initial conditions x(t 0 ) = x 0 = [ ] x1 (t 0 ) = x 2 (t 0 ) [ ] y(t0 ). (8) ẏ(t 0 ) The square matrix A R 2 2 is called the system dynamical matrix whereas B is the input matrix. Since y = x 1, then y = [ 1 0 ] x = Cx = x 1, (9) }{{} C where C is the output matrix. 7 / 14

8 State space representation The plant P, described by the second order differential equation ÿ = u M, has been transformed into 1 a first order matrix differential equation { ẋ = Ax + Bu, P := y = Cx + Du, where x R n, y R q, u R p, A R n n, B R n p, C R q n and D R q p is referred to as the state space representation of the dynamical system P. Note that this representation is not unique. 1 To consider the general case the input-output matrix D, now equal to zero, has been added. 8 / 14

9 Discrete time example The same reasonings apply also to discrete time system where t Z. Consider the Fibonacci sequence of numbers {0, 1, 1, 2, 3, 5,... } described by the recursive formula (or difference equation) with z(0) = 0 and z(1) = 1 and t 2. Then, defining z(t) = z(t 1) + z(t 2), (10) x(t) = [ ] x1 (t) x 2 (t) [ ] z(t 2) z(t 1) the second order difference equation becomes the first order matrix difference equation given by [ ] [ ] [ ] [ ] x1 (t + 1) z(t 1) x x(t + 1) = = = 2 (t) 0 1 = x(t) = Ax(t). x 2 (t + 1) z(t) x 1 (t) + x 2 (t) 1 1 }{{} A (11) With the initial condition x(0) = [0, 1], then y(t) = x 1 (t) is exactly the Fiboncacci series (how come?). In this specific example the system evolution does not depend on the input: the system is said to be autonomous. 9 / 14

10 Time-related properties Linearity Time invariant Linear Time Invariant Systems The system P is said to be causal if y(t) does not depend on future values of the input u(t), i.e. y(t) depends on u(τ) with τ t non-causal if y(t) depends also by future values of the input u(τ) with τ > t without memory if y(t) is unequivocally determined by the instantaneous value of u(t) with memory if y(t) depends on u(τ) for τ t. Examples / 14

11 Linearity: a key property Linearity Time invariant Linear Time Invariant Systems Using the functional relation P { } to compactly relate the system state x(t) (consequently its output y(t)) and its input u(t) such as then a system P is said to be additive if homegenous if x(t) = P {x(t 0 ), u(t)}, t t 0, x a(t) = P {x a(t 0 ), u a( )} x b (t) = P {x b (t 0 ), u b ( )} (implies) P {x a(t 0 )+x b (t 0 ), u a(t) + u b (t)} = x a(t) + x b (t). x(t) = P {x(t 0 ), u(t)}, (implies) P {αx(t 0 ), αu(t)} = αx(t). α R 11 / 14

12 Linearity: a key property cont d Linearity Time invariant Linear Time Invariant Systems additive homegenous P is Linear := additive + homogeneus if x a(t) = P {x a(t 0 ), u a(t)} x b (t) = P {x b (t 0 ), u b (t)} α R, β R P {αx a(t 0 )+βx b (t 0 ), αu a(t) + βu b (t)} = αx a(t) + βx b (t). Graphical examples / 14

13 Time invariant Linearity Time invariant Linear Time Invariant Systems A system P is said to be time invariant if the state evolution does not depend explicitly on time, i.e. x(t) = P {x(t 0 ), u(t T )} T 0, P {x(t 0 T ), u(t T )} = x(t T ). Examples / 14

14 LTI systems Linearity Time invariant Linear Time Invariant Systems The general state space representation of a linear time invariant (LTI) system is { ẋ = Ax + Bu, P := y = Cx + Du, in continuous time, and { x P := + = Ax + Bu, y = Cx + Du, in discrete time, where the step-forward or jump operator ( ) + is such that Prove that it is linear and time-invariant... x + (t) x(t + 1), t Z. 14 / 14