Continuous-time systems 2 March 3, 2015 Continuous-time systems 2 - - PowerPoint PPT Presentation

Properties of state-space representation Transfer functions

Continuous-time systems 2

March 3, 2015

Continuous-time systems 2

Properties of state-space representation Transfer functions

1

Properties of state-space representation

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Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Continuous-time systems 2

Properties of state-space representation Transfer functions

Outline

1

Properties of state-space representation

2

Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Continuous-time systems 2

Properties of state-space representation Transfer functions

Observability

A measure of how well a system’s internal states x can be inferred by knowledge of its outputs y.

Continuous-time systems 2

Properties of state-space representation Transfer functions

Observability

A measure of how well a system’s internal states x can be inferred by knowledge of its outputs y. Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs.

Continuous-time systems 2

Properties of state-space representation Transfer functions

Observability

A measure of how well a system’s internal states x can be inferred by knowledge of its outputs y. Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs. This holds for linear, time-invariant systems with n states if: rank(O) = n, O =      C CA . . . CAn−1      , O : observability matrix

Continuous-time systems 2

Properties of state-space representation Transfer functions

Controllability

A measure of the ability to move a system around in its entire configuration space using only certain admissible manipulations.

Continuous-time systems 2

Properties of state-space representation Transfer functions

Controllability

A measure of the ability to move a system around in its entire configuration space using only certain admissible manipulations. A system is controllable if its state can be moved from any initial state x0 to any final state xf via some finite sequence of inputs

• u0. . . uf .

Continuous-time systems 2

Properties of state-space representation Transfer functions

Controllability

A measure of the ability to move a system around in its entire configuration space using only certain admissible manipulations. A system is controllable if its state can be moved from any initial state x0 to any final state xf via some finite sequence of inputs

• u0. . . uf .

A linear, time-invariant system with n states is controllable if: rank(C) = n, C =

• B

AB . . . An−1B

• ,

where C is called the controllability matrix.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Outline

1

Properties of state-space representation

2

Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Transfer function

The transfer function of input i to output j is defined as: Hi,j(s) = Yj(s) Ui(s), U(s) = L{u(t)}, Y(s) = L{y(t)}. MIMO systems with n inputs and m outputs have n × m transfer functions, one for each input-output pair.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Transfer function

The transfer function of input i to output j is defined as: Hi,j(s) = Yj(s) Ui(s), U(s) = L{u(t)}, Y(s) = L{y(t)}. MIMO systems with n inputs and m outputs have n × m transfer functions, one for each input-output pair. The complex Laplace variable can be rewritten: s = σ + jω.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Transfer function

The transfer function of input i to output j is defined as: Hi,j(s) = Yj(s) Ui(s), U(s) = L{u(t)}, Y(s) = L{y(t)}. MIMO systems with n inputs and m outputs have n × m transfer functions, one for each input-output pair. The complex Laplace variable can be rewritten: s = σ + jω. The frequency response of a system can be analyzed via H(jω): eσ+jω = eσ(cos ω + j sin ω).

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Illustration of Euler’s formula

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Poles and zeros

In general, the transfer function can be written as: H(s) = N(s) D(s).

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Poles and zeros

In general, the transfer function can be written as: H(s) = N(s) D(s). The poles of H(s) are zeros of D(s), ie {s : D(s) = 0}. |H(s)| = ∞ if s is a pole.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Poles and zeros

In general, the transfer function can be written as: H(s) = N(s) D(s). The poles of H(s) are zeros of D(s), ie {s : D(s) = 0}. |H(s)| = ∞ if s is a pole. The zeros of H(s) are zeros of N(s), ie {s : N(s) = 0}. H(s) = 0 if s is a zero.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Poles and zeros

In general, the transfer function can be written as: H(s) = N(s) D(s). The poles of H(s) are zeros of D(s), ie {s : D(s) = 0}. |H(s)| = ∞ if s is a pole. The zeros of H(s) are zeros of N(s), ie {s : N(s) = 0}. H(s) = 0 if s is a zero. Poles and zeros may cancel, ie. if D(s) = N(s) = 0 for some s.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Steady-state response

The output of a linear time-invariant system yields consists of: a steady-state output yss(t), which similar periodicity to u(t) → yss comprises the same frequencies as u(t) a transient output ytr(t) → if the system is stable, then limt→∞ ytr(t) = 0 → ytr(t) depends on the initial state x0(t) of the system

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Steady-state response

The output of a linear time-invariant system yields consists of: a steady-state output yss(t), which similar periodicity to u(t) → yss comprises the same frequencies as u(t) a transient output ytr(t) → if the system is stable, then limt→∞ ytr(t) = 0 → ytr(t) depends on the initial state x0(t) of the system If we apply an input u(t) = cos(αt + θ), then: yss(t) = |H(jα)|cos(αt + θ + ∠H(jα))

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Steady-state response

The output of a linear time-invariant system yields consists of: a steady-state output yss(t), which similar periodicity to u(t) → yss comprises the same frequencies as u(t) a transient output ytr(t) → if the system is stable, then limt→∞ ytr(t) = 0 → ytr(t) depends on the initial state x0(t) of the system If we apply an input u(t) = cos(αt + θ), then: yss(t) = |H(jα)|cos(αt + θ + ∠H(jα)) The steady-state output yss(t) of a linear time invariant system: consists of signals of same frequencies as the input signal u(t) which may have been magnified and/or phase changed

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Outline

1

Properties of state-space representation

2

Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Impulse response

The impulse response h(t) of input i to output j is the output yj(t) of a system when an impulse δ(t) is applied at input ui(t).

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Impulse response

The impulse response h(t) of input i to output j is the output yj(t) of a system when an impulse δ(t) is applied at input ui(t). The impulse response is the inverse Laplace transform of the transfer function h(t) = L−1{H(s)}.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Impulse response

The impulse response h(t) of input i to output j is the output yj(t) of a system when an impulse δ(t) is applied at input ui(t). The impulse response is the inverse Laplace transform of the transfer function h(t) = L−1{H(s)}. For stable continuous time systems the impulse response always converges to 0: lim

t→∞ h(t) = 0, because D = 0 and

lim

t→∞ x(t) = 0.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Impulse response

The impulse response h(t) of input i to output j is the output yj(t) of a system when an impulse δ(t) is applied at input ui(t). The impulse response is the inverse Laplace transform of the transfer function h(t) = L−1{H(s)}. For stable continuous time systems the impulse response always converges to 0: lim

t→∞ h(t) = 0, because D = 0 and

lim

t→∞ x(t) = 0.

The speed of convergence depends on the position of the poles.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Time constant

The transfer function of first order systems can be written as: H(s) = K τs + 1 ↔ h(t) = K τ e−t/τ, where τ is called the system’s time constant.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Time constant

The transfer function of first order systems can be written as: H(s) = K τs + 1 ↔ h(t) = K τ e−t/τ, where τ is called the system’s time constant. The time constant summarizes the speed of a system’s dynamics: after τ seconds, the impulse response reaches h(0)/e. after τ seconds, the step response has reached 1 − e−1 ≈ 63%

• f its regime value.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Impulse response H(s) = 5/(5s + 1) ↔ h(t) = exp(−t/5)

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Step response H(s) = 5/(5s + 1) ↔ h(t) = exp(−t/5)

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Outline

1

Properties of state-space representation

2

Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

From state-space to transfer functions

We start from the linear state-space representation: time domain Laplace domain ˙ x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) ↔ sX(s) = AX(s) + BU(s) Y(s) = CX(s) + DU(s)

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

From state-space to transfer functions

We start from the linear state-space representation: time domain Laplace domain ˙ x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) ↔ sX(s) = AX(s) + BU(s) Y(s) = CX(s) + DU(s) A transfer function H(s) = Y(s)

U(s) relates an input and an output in

the Laplace-domain → to obtain it, we must eliminate X(s).

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

From state-space to transfer functions

We start from the linear state-space representation: time domain Laplace domain ˙ x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) ↔ sX(s) = AX(s) + BU(s) Y(s) = CX(s) + DU(s) A transfer function H(s) = Y(s)

U(s) relates an input and an output in

the Laplace-domain → to obtain it, we must eliminate X(s). (sI − A)X(s) = BU(s) X(s) = (sI − A)−1BU(s) ⇒ Y(s) = C(sI − A)−1BU(s) + DU(s) ⇒ H(s) = C(sI − A)−1B + D

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 1/2

Poles are zeros of the denominator of H(s), e.g. those values of s for which H(s) is singular.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 1/2

Poles are zeros of the denominator of H(s), e.g. those values of s for which H(s) is singular. The relationship between state-space representation (matrices A, B, C and D) and transfer functions is given by H(s) = C(sI − A)−1B + D

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 1/2

Poles are zeros of the denominator of H(s), e.g. those values of s for which H(s) is singular. The relationship between state-space representation (matrices A, B, C and D) and transfer functions is given by H(s) = C(sI − A)−1B + D H(s) cannot be computed when (sI − A)−1 does not exist, ie. det(sI − A) = 0

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 1/2

Poles are zeros of the denominator of H(s), e.g. those values of s for which H(s) is singular. The relationship between state-space representation (matrices A, B, C and D) and transfer functions is given by H(s) = C(sI − A)−1B + D H(s) cannot be computed when (sI − A)−1 does not exist, ie. det(sI − A) = 0 The determinant is zero if s is an eigenvalue of A.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 1/2

Poles are zeros of the denominator of H(s), e.g. those values of s for which H(s) is singular. The relationship between state-space representation (matrices A, B, C and D) and transfer functions is given by H(s) = C(sI − A)−1B + D H(s) cannot be computed when (sI − A)−1 does not exist, ie. det(sI − A) = 0 The determinant is zero if s is an eigenvalue of A. → all poles of H(s) are eigenvalues of A

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 2/2

Transfer functions only capture what is relevant to describe an input-output relationship, but not all states necessarily contribute.

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 2/2

Transfer functions only capture what is relevant to describe an input-output relationship, but not all states necessarily contribute. → unobservable modes of A are not poles in H(s).

Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 2/2

Transfer functions only capture what is relevant to describe an input-output relationship, but not all states necessarily contribute. → unobservable modes of A are not poles in H(s). Consider the following SISO system with 2 states: sX1(s) sX2(s)

• =

α 0.2 1 X1(s) X2(s)

• +

β 2

• U(s)

Y (s) =

• 1

X1(s) X2(s)

• Continuous-time systems 2

Properties of state-space representation Transfer functions Impulse response and time constant Relationship between state space and transfer functions

Relationship between poles and eigenvalues of A 2/2

Transfer functions only capture what is relevant to describe an input-output relationship, but not all states necessarily contribute. → unobservable modes of A are not poles in H(s). Consider the following SISO system with 2 states: sX1(s) sX2(s)

• =

α 0.2 1 X1(s) X2(s)

• +

β 2

• U(s)

Y (s) =

• 1

X1(s) X2(s)

• The transfer function H(s) =

β s−α has only one pole (s1 = α).

→ not all eigenvalues of A are poles in transfer functions H(s).

Continuous-time systems 2