Introduction to Computer Control SystemsLecture 6: LTI system response
Dave Zachariah
Div. Systems and Control, Dept. Information Technology,Uppsala University
December 9, 2014
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 1 / 12
Today’s lecture: What and why?
LTI system responsesWhy: Characterize what your system does to a well-defined inputsignal. Control design criteria are often defined by using specificationsof the step response.
Observability and controllabilityWhy: Is the system model such that we can observe all state changesthrough the output signal? Can we affect all the states using our inputsignal?
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 2 / 12
Today’s lecture: What and why?
LTI system responsesWhy: Characterize what your system does to a well-defined inputsignal. Control design criteria are often defined by using specificationsof the step response.Observability and controllabilityWhy: Is the system model such that we can observe all state changesthrough the output signal? Can we affect all the states using our inputsignal?
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 2 / 12
System example: Spring with input force
LTI system in state-space form:
x = Ax + Bu
y = Cx +Du
y
u
States and output:
x =
[x1x2
]=
[yy
]y =
[1 0
]x
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 3 / 12
System example: Spring with input force
Input u(t) is an impulse δ(t)
0 20 40 60 80 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
t [s]
inp
ut
u(t
), o
utp
ut
y(t
)
u(t)
y(t)
Temporal perspective
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 4 / 12
System example: Spring with input force
Input u(t) is an impulse δ(t)
−1 −0.5 0 0.5 10
0.02
0.04
0.06
0.08
0.1
x1 (position) [m]
x2 (
ve
locity)
[m/s
]
State-space perspective x(t) at t = 0+.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 4 / 12
System example: Spring with input force
Input u(t) is an impulse δ(t)
0 0.05 0.1 0.15 0.2 0.25 0.3−0.02
0
0.02
0.04
0.06
0.08
0.1
x1 (position) [m]
x2 (
ve
locity)
[m/s
]
State-space perspective x(t) at t = 5.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 4 / 12
System example: Spring with input force
Input u(t) is an impulse δ(t)
−0.2 −0.1 0 0.1 0.2 0.3
−0.05
0
0.05
x1 (position) [m]
x2 (
ve
locity)
[m/s
]
State-space perspective x(t) at t = 20.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 4 / 12
System example: Spring with input force
Input u(t) is an impulse δ(t)
−0.2 −0.1 0 0.1 0.2 0.3
−0.05
0
0.05
x1 (position) [m]
x2 (
ve
locity)
[m/s
]
State-space perspective x(t) at t = 100.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 4 / 12
LTI system response: transient and steady-state
y(t)u(t)G
LTI system in state-space form
x = Ax + Bu
y = Cx +Du
Assume x(0) = 0 then
y(t) =
∫ t
τ=0CeA(t−τ)Bu(τ)dτ +Du(t)
= dinput is a step u(t) = 1 for t ≥ 0e= CA−1eAtB︸ ︷︷ ︸
transient response
+ −CA−1B +D︸ ︷︷ ︸steady-state response
, t ≥ 0
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 5 / 12
LTI system response: transient and steady-state
y(t)u(t)G
LTI system in state-space form
x = Ax + Bu
y = Cx +Du
Assume x(0) = 0 then
y(t) =
∫ t
τ=0CeA(t−τ)Bu(τ)dτ +Du(t)
= dinput is a step u(t) = 1 for t ≥ 0e= CA−1eAtB︸ ︷︷ ︸
transient response
+ −CA−1B +D︸ ︷︷ ︸steady-state response
, t ≥ 0
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 5 / 12
LTI system response: transient and steady-state
Input u(t) is a unit step (u(t) = 1 for t ≥ 0.)
0 20 40 60 800
5
10
15
t [s]
inp
ut
u(t
), o
utp
ut
y(t
)
u(t)
y(t)
Temporal perspective
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 6 / 12
LTI system response: transient and steady-state
Input u(t) is a unit step (u(t) = 1 for t ≥ 0.)
0 20 40 60 800
5
10
15
t [s]
inp
ut
u(t
), o
utp
ut
y(t
)
u(t)
y(t)
Steady-state yss (recall final value theorem) and overshoot M .
Rise time Tr: time it take for y(t) to go from 0.1yss to 0.9yss.
Settling time T ps : time it takes for y(t) to stay within (1± p)yss.(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 6 / 12
LTI system response: transient and steady-state
Input u(t) is a unit step (u(t) = 1 for t ≥ 0.)
0 20 40 60 800
5
10
15
t [s]
inp
ut
u(t
), o
utp
ut
y(t
)
u(t)
y(t)
Steady-state yss (recall final value theorem) and overshoot M .
Rise time Tr: time it take for y(t) to go from 0.1yss to 0.9yss.
Settling time T ps : time it takes for y(t) to stay within (1± p)yss.(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 6 / 12
LTI system response: transient and steady-state
Input u(t) is a unit step (u(t) = 1 for t ≥ 0.)
0 5 10 15−2
−1
0
1
2
3
x1 (position) [m]
x2 (
ve
locity)
[m/s
]
State-space perspective x(t) at t = 100.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 6 / 12
LTI system response: transient and steady-state
Input u(t)= cos(ωt) for t ≥ 0.
0 20 40 60 80 100−6
−4
−2
0
2
4
t [s]
inp
ut
u(t
), o
utp
ut
y(t
)
u(t)
y(t)
Temporal: Note transient vs. stationary/steady-state of y(t)
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 7 / 12
LTI system response: transient and steady-state
Input u(t)= cos(ωt) for t ≥ 0.
−6 −4 −2 0 2 4−3
−2
−1
0
1
2
3
x1 (position) [m]
x2 (
ve
locity)
[m/s
]
State-space perspective x(t) at t = 100.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 7 / 12
Controllability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
A particular state x? is controllable if we can apply an input u(t)that takes the system from x(0) = 0 to x? in finite time T .On the board: Illustrate
Recall solution of state
x(T ) =
∫ T
t=0eA(T−τ)Bu(τ)dτ
= dusing Cayley-Hamilton’s theorem we get following forme= Bγ0 + ABγ1 + · · ·+ An−1Bγn−1
⇒ a given state x(T ) is linear combination of B,AB, · · · ,An−1B.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 8 / 12
Controllability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
A particular state x? is controllable if we can apply an input u(t)that takes the system from x(0) = 0 to x? in finite time T .On the board: Illustrate
Recall solution of state
x(T ) =
∫ T
t=0eA(T−τ)Bu(τ)dτ
= dusing Cayley-Hamilton’s theorem we get following forme= Bγ0 + ABγ1 + · · ·+ An−1Bγn−1
⇒ a given state x(T ) is linear combination of B,AB, · · · ,An−1B.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 8 / 12
Controllability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
A particular state x? is controllable if we can apply an input u(t)that takes the system from x(0) = 0 to x? in finite time T .On the board: Illustrate
Recall solution of state
x(T ) =
∫ T
t=0eA(T−τ)Bu(τ)dτ
= dusing Cayley-Hamilton’s theorem we get following forme= Bγ0 + ABγ1 + · · ·+ An−1Bγn−1
⇒ a given state x(T ) is linear combination of B,AB, · · · ,An−1B.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 8 / 12
Controllability of LTI systems, cont’d
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
⇒ a given state x(T ) is linear combination of B,AB, · · · ,An−1B.
All states x? are controllable if and only if matrix
S(A,B) =[B AB · · · An−1B
]has n independent columns. That is, S has full rank.
System G is controllable ⇔ all x? are controllable ⇔ rank(S) = n.
Important property for designing controllers.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 9 / 12
Controllability of LTI systems, cont’d
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
⇒ a given state x(T ) is linear combination of B,AB, · · · ,An−1B.
All states x? are controllable if and only if matrix
S(A,B) =[B AB · · · An−1B
]has n independent columns. That is, S has full rank.
System G is controllable ⇔ all x? are controllable ⇔ rank(S) = n.
Important property for designing controllers.
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 9 / 12
Observability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
Suppose we have zero input u(t) ≡ 0 and initialize system at somestate x(0) = x? 6= 0. Then x? is unobservable if output isunchanged y(t) ≡ 0.On the board: Illustrate
If output signal is constant y(t) = 0, then all derivatives at t = 0are
dk
dtky(t)|t=0 = C
dkx(t)
dtk|t=0=CAkx?= 0
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 10 / 12
Observability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
Suppose we have zero input u(t) ≡ 0 and initialize system at somestate x(0) = x? 6= 0. Then x? is unobservable if output isunchanged y(t) ≡ 0.On the board: Illustrate
If output signal is constant y(t) = 0, then all derivatives at t = 0are
dk
dtky(t)|t=0 = C
dkx(t)
dtk|t=0=CAkx?= 0
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 10 / 12
Observability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
Constant y(t) = 0, means that for observable x? 6= 0 we have
dk
dtky(t)|t=0=CAkx? 6= 0
⇒ Cx? 6= 0, CAx? 6= 0, · · · , CAn−1x? 6= 0
That is, x? 6= 0 is observable if
O(C,A) =
CCA
...CAn−1
x? 6= 0
System G is observable ⇔ all x? 6= 0 are observable ⇔rank(O) = n. (So, Ox? = 0 impossible for x? 6= 0)
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 11 / 12
Observability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
Constant y(t) = 0, means that for observable x? 6= 0 we have
dk
dtky(t)|t=0=CAkx? 6= 0
⇒ Cx? 6= 0, CAx? 6= 0, · · · , CAn−1x? 6= 0
That is, x? 6= 0 is observable if
O(C,A) =
CCA
...CAn−1
x? 6= 0
System G is observable ⇔ all x? 6= 0 are observable ⇔rank(O) = n. (So, Ox? = 0 impossible for x? 6= 0)
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 11 / 12
Observability of LTI systems
LTI system of order n in state-space form
x = Ax + Bu
y = Cx +Du.
Constant y(t) = 0, means that for observable x? 6= 0 we have
dk
dtky(t)|t=0=CAkx? 6= 0
⇒ Cx? 6= 0, CAx? 6= 0, · · · , CAn−1x? 6= 0
That is, x? 6= 0 is observable if
O(C,A) =
CCA
...CAn−1
x? 6= 0
System G is observable ⇔ all x? 6= 0 are observable ⇔rank(O) = n. (So, Ox? = 0 impossible for x? 6= 0)
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 11 / 12
Today’s lecture: What and why?
LTI system responsesWhy: Characterize what your system does to a well-defined inputsignal. Control design criteria are often defined by using specificationsof the step response.Observability and controllabilityWhy: Is the system model such that we can observe all state changesthrough the output signal? Can we affect all the states using our inputsignal?
(UU/Info Technology/SysCon) Intro. Computer Control Sys. December 9, 2014 12 / 12