Forced response and Initial-Condition Response

Systems and Control Theory Lecture Notes

Laura Giarr´ e

Lesson 6: Response of LTI Systems

 Forced response and Initial-Condition Response

 DT Initial condition response

 DT Forced response

 CT Initial condition response

 CT Forced response

Forced response and initial condition response 1

 We want to study the output of a system starting at time t ₀ , knowing the initial state x(t ₀ ) = x 0 , and the present and

future input u(t), t ≥ t ₀ . Let us study the following two cases:

 Initial-conditions response:

 x _IC (t ₀ ) = x ₀ ,

u _IC (t) = 0, t ≥ t 0 , → y _IC

 Forced response:

 x _F (t ₀ ) = 0,

u _F (t) = u(t), t ≥ t ₀ , → y _F .

Forced response and initial condition response 2

 Clearly, x ₀ = x _IC + x _F , and u = u _IC + u _F , hence y = y _IC + y _F

 We can always compute the output of a linear system by

adding the output corresponding to zero input and the original initial conditions, and the output corresponding to a zero

initial condition, and the original input.

 We can study separately the effects of non-zero inputs and of non-zero initial conditions.

 The complete case can be recovered from these two.

DT: Initial conditions response

 Consider the case of zero input, i.e., u = 0; in this case, the state-space equations are written as the difference equations:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

x[0] = x ₀ y[0] = C [0]x ₀

x[1] = A[0]x[0] y[1] = C [1]A[0]x[0]

x[2] = A[1]A[0]x[0] y[2] = C [2]A[1]A[0]x[0]

... ...

x[k] = Φ[k, 0]x[0] y[k] = C [k]Φ[k, 0]x[0]

where we defined the state transition matrix Φ[k, ]as Φ[k, ] =

 A[k − 1]A[k − 2]...A[], k >  ≥ 0

I, k = 

Forced response with zero i.c. (DT) 1

 We need to compute the solution of x[k + 1] = Ax[k] + Bu[k], x[0] = 0.

 By substitution, we get:

x[k] =A[k − 1]x[k − 1] + B[k − 1]u[k − 1]

=A[k − 1](A[k − 2]x[k − 2] + B[k − 1]u[k − 2]) + B[k − 1]u[k − 1]

= Φ[k, 0]x[0]  

=0

+

k−1 

i=0

Φ[k, i + 1]B[i]u[i]

Forced response with zero i.c. (DT) 2

 In other words,

x[k] = Γ[k, 0]U[k, 0]

 where

Γ[k, 0] = Φ[k, 1]B[0]Φ[k, 2]B[1]...B[k − 1]

U =

⎡

⎢ ⎢

⎣

u[0]

u[1] ...

u[k − 1]

⎤

⎥ ⎥

⎦

 The output is

y[k] = C [k]Γ[k, 0]U[k, 0].

DT Response

 In general, state/output trajectories of a DT state-space model can be computed as:

x[k] =Φ[k, 0]x[0] + Γ[k, 0]U[k, 0]

y[k] =C [k]Φ[k, 0]x[0] + C [k]Γ[k, 0]U[k, 0]

 In general Φ[k, ] may not be invertible. In the cases in which

it is, one can also compute x[0] as a function of x[k].

Initial-Condition Response (CT) 1

 Consider the case of zero input u = 0, then d

dt x(t) =A(t)x(t), x(t ₀ ) = x(0) y(t) =C (t)x(t);

 Assume that the matrix function A(t) is sufficiently well behaved (e.g. piecewise continuous) so that there exists unique state/output signals x and y.

 Define a state transition function Φ(t, τ) such that, for all t, τ ∈ T,

∂

∂t Φ(t, τ) =AΦ(t, τ))

Φ(t, t) =I;

Initial-Condition Response (CT) 2

 The function Φ can in general be computed numerically, integrating a differential equation in n unknown functions, with n initial conditions ( x ∈ R ⁿ ).

 Then,

x(t) = Φ(t, t ₀ )x 0

y(t) = C (t)Φ(t, t ₀ )x ₀

Forced Response with zero i.c. (CT) 1

 We need to integrate d

dt x(t) =A(t)x(t) + B(t)u(t), x(t ₀ ) = 0 y(t) =C (t)x(t) + D(t)u(t);

 Again, assume that the input signal u and the matrix

functions A and B are such that there exists a unique solution.

Claim: the forced solution is x(t) =

 _t

t

Φ(t, τ)B(τ)u(τ)dτ.

y(t) = C (t)

 _t

Φ(t, τ)B(τ)u(t)dτ + D(t)u(t)

Forced Response with zero i.c. (CT) 2

 Proof: Verify by substitution clearly x(t0) = 0; moreover, d

dt x(t) = d dt

 _t

t

Φ(t, τ)B(τ)u(τ)dτ

=

 _t

t

∂

∂t Φ(t, τ)B(τ)u(τ)dτ + [Φ(t, τ)B(τ)u(τ)dτ] τ=t

= A(t)

 _t

t

Φ(t, τ)B(τ)u(τ)dτ + B(t)u(t)

= A(t)x(t) + B(t)u(t)



Properties of the state transition matrix

 Further properties of the state transition function

 Composition

Φ(t 2 , t 0 ) = Φ(t 2 , t 1 )Φ(t 1 , t 0 ).

 Jacobi-Liouville formula

det (Φ(t, t ₀ ) = exp(

 _t

t

trace[A(τ )]dτ

 Note that in the CT case the transition matrix is invertible,

but this is not always the case in DT case

Thanks