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CONTROL ROOM ACCELERATOR PHYSICS
Day 2 Introduction to Control Theory
Control Theory Introduction • Control theory is an interdisciplinary branch of engineering and mathematics
that deals with the behavior of dynamical systems with inputs. When one or more output variables of a system needs to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system.
• A typical objective of a control theory is to calculate proper corrective action for inputs that results in system stability, that is, the system will hold the set point and not oscillate around it.
• Branches of Control Theory • Adaptive control: Adapt process and gain parameters to observed behavior • Intelligent control: Application of AI techniques in controller design • Optimal control: Controller is designed to extremize a given objective • Robust control: Design controller to be insensitive to model errors (eg., H∞ control) • Stochastic control: Design control to be insensitive to model uncertainties • Energy shaping control: Plant and controller are viewed as energy transformers • Self-organized criticality control: Controller is aware of self-organizing systems • …
1/27/14 USPAS 2
Control Theory Introduction • Control theory is relatively young and a very active research area.
• Accelerator control utilizes very little formal control! Wide open.
• Control theory is technical. • Dynamical Systems • Abstract algebra • Functional analysis • Signal processing • Differential manifolds • Abstract algebra (groups, rings, modules, …) • Lie groups and algebras • Representation theory (representing algebraic structure with matrices)
• Applications in
• Aerospace • Automotive • Robotics • Biology • Economics
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Outline
1. Introduction
2. Linear dynamical systems
3. Modeling linear systems
4. Control theory: Stability, observability, controllability
5. Control examples: Perturbation and disturbance rejection
1/27/14 USPAS 4
We restrict our attention to the basic tenets of control theory and linear dynamical systems, with some examples found in accelerator control.
Linear Dynamical Systems Introduction
A Dynamical System is a system G that varies in time. Typically it has inputs u and outputs y. In control theory G is usually referred to as a plant.
Linear Dynamical Systems are dynamical
systems G where the input u and output y are linearly related • If u = u1 + u2 is the input then output y is
y = G(u1 + u2) = G(u1) + G(u2)
• Linear systems may be • Continuous time (ODEs) • Discrete time (delay equations) • Typically these are related!
1/27/14 USPAS 5
Plant G u
y
plant input
plant output
Dynamical System
kkkk uybybyb
tutybtybtyb
=++
=++
−− 01122
012 )()()()( continuous time
discrete time
Linear Dynamical Systems Example: Linear Beam Optics, a Discrete “Time” Case
1/27/14 USPAS 6
Beam
Beamline
Φ(u1)
Φ(u2)
Φ(un-1)
Φ(un-2)
Model of Beamline (cascade of transfer matrices) Dynamical System
z0(t)
System state (e.g., beam location)
Linear Dynamical Systems Note on Modeling Physics and Engineering Problems
• Most linear dynamical systems in physics and engineering are not naturally expressed in the form G(u) = y (output as function of input) • The output y is generally a combination of differentials of itself! • For example, consider the 2nd order linear operator L
• Then our model appears as
• This is a linear equation, but the wrong direction! By comparing equations we find (abstractly)
1/27/14 USPAS 7
)()()()( 012 tybtybtyby ++≡ L
uy =)(L
)()( 1 uLuGy −== What does L-1 mean?!
Linear Dynamical Systems The State Space Representation (Common in Control Theory) What is y = G(u) = L-1(u) for a linear differential system ? • Start with our 2nd order linear differential equation
• Define our state variables x1 and x2
• Differentiate ; then differentiate x2 yielding
• Arranging into matrix-vector format
1/27/14 USPAS 8
uybby
bbytutybtybtyb +−−==++
2
0
2
1012 or),()()()(
yxyx≡
≡
2
1
uxbbx
bbyx +−−== 1
2
02
2
12
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=⎟⎟⎠
⎞⎜⎜⎝
⎛
2
1
2
1
2
0
2
12
1
01
1010
xx
y
uxx
bb
bb
xx
dtd
x1 = y = x2
)()()()()(
ttttt
CxyBuAxx
=
+=This has form
output
Linear Time-Invariant Dynamical System State Space Representation
• The general representation of an nth-order Linear Time-Invariant (LTI) dynamical system is (for continuous case)
• A, B, C, D are matrices of the appropriate dimensions
• x is the state vector (plant internal dynamics – e.g., the beam) • y is the output vector (sensor output, what we can observe – e.g., BPM response) • u is the input vector (actuator input – e.g., magnet strengths)
• Often we drop the matrix D since we can renormalize output y
• The above is called the state space representation
1/27/14 USPAS 9
x(t) =Ax(t)+Bu(t), x(t)∈ Rn, y(t)∈ Rm, u(t)∈ R p
y(t) =Cx(t)+Du(t), A ∈ Rn×n, B∈ Rn×p, C∈ Rm×n, D∈ Rm×py = G(u)
¡ The matrices A, B, C determine plant properties l Matrix A determines the internal plant dynamics l Matrix B determines how input affects internal dynamics l Matrix C determines coupling between internal dynamics and output
LTI Dynamical System Block Diagram of State Space Representation
x
1/27/14 USPAS 10
Internal Dynamics (beam dynamics)
∫
A
x B
Actuators (correctors)
C
Sensors (BPMS)
u y input output
x =Ax+Buy =Cx
LTI Dynamical System Solution to the State Equations
Solution to state vector equations is
• This is completely analogous to the scalar case
• Internal dynamics x(t) are the sum of natural system response etA and the natural response convolved with the driving term Bu(t) • Note A is square and the matrix exponential function etA is well-defined
• Thus, the system dynamics are dictated by the matrix etA • The control properties are dictated by matrix B • The observation properties are dictated by matrix C
1/27/14 USPAS 11
x(t) =Ax(t)+Bu(t) A ∈ Rm×m, B∈ Rm×n
y(t) =Cx(t) C∈ Rk×m
x(t) = etAx0 + e(t−τ )ABu(τ )dτ0
t
∫y(t) =Cx(t)
x(t) = ax(t)+ bu(t), y(t) = cx(t)
LTI Dynamical System Discrete Case
• For discrete case the state representation looks like
• The solution to the state variable equation is
• This solutions is analogous to the continuous case, only the natural response is dictated directly by matrix Ak rather than etA, as we shall see.
1/27/14 USPAS 12
kk
kkk
CxyBuAxx
=
+=+1
∑−
=
−−+=1
0
10
k
ik
ikkk BuAxAx
0 0
uqh (ush1,usv1) uqv (ush2,usv2)
(x,y,z)Qh S1 S2Qh Qv QvD1
D2
LTI Dynamical System Example: Discrete State Space Modeling Beam Steering
Say we have Beam Position Monitors (BPMs) as our sensors, then our observables are the coordinates (x,y,z); that is, we do not have access to the full state vector – no momentum components.
Note state vector z and observation vector y • State vector z = (x, x’,y,y’,z,z’)
• Observation vector
• Then
• where
• With {Φk} as the transfer matrices, our modeling equations are
1/27/14 USPAS 13
( )Tzyx≡y
yk =Czk (t)
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
≡
010000000100000001
C
zk+1 =Φk (uk )zkyk =Czk
These are in the form of the discrete state space representation
Control Theory Application of Control Theory to Dynamical Systems OUTLINE
• Dynamical System • Stability • Controllability • Observability
• Control • Closed-loop/Open-loop • Regulator • State Observer • PID Control
1/27/14 USPAS 14
Control Theory Stability
• Loosely, the stability of a dynamical system at equilibrium point y0 indicates that the output y(t) always “stays near” y0 for sufficiently small inputs u(t).
• A linear system is stable if all bounded inputs u produce bounded outputs y (||u(t)|| < ∞ => ||y(t)|| < ∞ all t)
• If the system eventually returns to y0, it is said to be asymptotically stable: the variables of an asymptotically stable control system do not permanently oscillate.
• If the response to an input u(t) neither decays nor grows over time, it is marginally stable. The output may oscillate indefinitely about y0 but will not blow up.
• Otherwise the system is unstable.
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See http://vimeo.com/2952236
• Examples • An ideal pendulum is marginally stable at y0 = 0˚ • A damped pendulum is asymptotically stable at y0 = 0˚ • An (inverted) pendulum is unstable at y0 = 180˚
• However, we can stabilize an inverted pendulum with
active stabilization u(t) = u[y(·),t] • “Fly by wire” • (Stability is the opposite of maneuverability)
Control Theory Controllability
Roughly speaking, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible inputs. The exact definition varies slightly within the framework or the type of models applied.
For us, controllability means there exists at least one valid control strategy u(·) : R+ → Rp that brings the system to any output y ∈ Rm.
1/27/14 USPAS 16
Is the Rubik’s cube controllable?
G(u)
y(t)
u(t) y
Control Theory Observability
• Informally, observability is a measure of how well the internal states x of a system can be inferred by knowledge of its external outputs y.
• More formally, a system is said to be observable if, for any possible sequence of control vectors u(·) we can determine the current state x(t) in finite time t < ∞ only by watching the outputs y(·). Note we know both u(t) and y(t) for all t.
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• If a system is not observable, this means the current values of some of its states cannot be determined through output sensors. Their value is unknown to the controller (although they might be estimated through various means).
• The observability and controllability of a system are mathematical duals.
• The concept of observability was introduced by American-Hungarian scientist Rudolf E. Kalman for linear dynamic systems.
¡ The matrices A, B, C determine plant properties l Matrix A determines stability (we cover this) l Matrices A and B determine controllability (outputs we can reach) l Matrices A and C determine observability (watching the output says?)
Internal Dynamics (beam dynamics)
Control Theory Stability, Controllability, and Observability of an LTI System
x
1/27/14 USPAS 18
∫
A
B u y
C
x
Actuators (correctors)
Sensors (BPMS)
input output
x =Ax+Buy =Cx
• Consider the scalar case • x(t), y(t), a, b, c ∈ R
• “Solution” to is
• Differentiate to prove it (Exercise) • Internal dynamics are the sum of natural response eat x0 at initial condition x0,
plus the natural response convolved (“folded”) with the driving term bu(t).
• Finally, system solution y(t) is proportional to x(t),
Control Theory An Example LTI System: Continuous Scalar System
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x(t) = ax(t)+ bu(t)y(t) = cx(t)
)()()( tbutaxtx +=
x(t) = eat x0 + ea(t−τ )bu(τ )dτ0
t
∫
y t( ) = cx(t) = ceat x0 + c ea(t−τ )bu(τ )dτ0
t
∫
= ceat x0 + b e−aτu(τ )dτ0
t
∫#
$%
&
'(
Observability • So long as c ≠ 0
Controllability • So long as b ≠ 0
Infer about Matrix-Vector Systems • Look at eigenvalues/singular values of A, B, C
• Positive eigenvalues of A indicate instability • Zero singular values of B indicate uncontrollable modes • Zero singular values of C indicate unobservable modes
Stability
Control Theory Stability, Controllability, and Observability of Example System System
With solution
• For natural response [eat term] • a > 0 eat unbounded • a = 0 eat is stable • a < 0 eat decays
1/27/14 USPAS
20
cxybuaxx
=
+=
Unstable Asymptotically stable
Marginally stable
y(t) = ceat x0 + c ea(t−τ )bu(τ )dτ0
t
∫
a 0
Scalar Stability Region
Stability with Inputs • Consider driven response (u ≠ 0)
• Note (t - τ) > 0 so ea(t - τ) amplifies or attenuates u(t) according to whether a > 0 or a < 0, respectively.
• Thus, system is stable iff a ≤ 0. • Stability dominated by natural response eat
eatx0
x0
Stable system iff a ≤ 0
Control Theory Stability, Controllability, and Observability of the General LTI System
Solution to vector equations is
• This is completely analogous to the scalar case
• Again, internal dynamics are the sum of natural response etA plus the natural response convolved with the driving term Bu(t) • Note A is square and the matrix exponential function etA is well-defined
• The state dynamics are dictated by the matrix H(t) = etA • The control dynamics are dictated by matrix B • The observation dynamics are dictated by matrix C
• Positive eigenvalues of A indicate instability • Zero singular values of B indicate uncontrollable modes • Zero singular values of C indicate unobservable modes
1/27/14 USPAS 21
x(t) =Ax(t)+Bu(t) A ∈ Rm×m, B∈ Rm×n
y(t) =Cx(t) C∈ Rk×m
y(t) =CetAx0 +C e(t−τ )ABu(τ )dτ0
t
∫
Unstable: exponentially
increasing
Control Theory Stability of Continuous LTI Systems
• Let λ be an eigenvalue of A • Dynamics of eigenmode eλ are determined by etλ
• λ on the complex plane
then
• A finite imaginary part of λ (ω≠0) implies oscillation • A zero imaginary part of λ (ω=0) implies none
• A negative real part of λ (σ<0) indicates exponential
decay – asymptotically stable
• A positive real part of λ (σ>0) indicates exponential growth – unstable
1/27/14 USPAS 22
Re λ = σ
Im λ = ω
Marginally stable, oscillatory
Asymtotically stable: exponentially
decreasing
Non-oscillatory Let λ =σ + jω σ ,ω ∈ R
etλ = eσ t cosωt + ieσ t sinωt
Control Theory Controllability of LTI Systems • The LTI state solution
• Recall system is controllable iff we can steer to any output η ∈ Rn • There exist a u1(•) and t1 > 0 such that y[t1;u1(t1)] = η .
• LTI controllability criteria: 1. Matrix etAB has full row rank (true?) 2. Augmented matrix [B | AB | … | An-1B] has full row rank
• That is, there are n linearly independent columns 3. Show 2. implies 1. (hint, expand etA)
1/27/14 USPAS 23
x(t) = etAx0 + e(t−τ )ABu(τ )dτ0
t
∫
Control Theory Observability of an LTI System • The LTI output solution
• Recall system is observable iff we can deduce the system state x ∈ Rm by observing input u(•) and output y(•) for finite t>0.
• LTI observability criteria: 1. Matrix CetA has full row rank 2. Augmented matrix [CT | (CA)T | … | (CAk-1) T]T has full row rank
• That is, there are n linearly independent columns 3. Show 2. implies 3. (hint, expand etA)
1/27/14 USPAS 24
x(t) = etAx0 + e(t−τ )ABu(τ )dτ0
t
∫y(t) =Cx(t)
CCA
CAk−1
"
#
$$$$$
%
&
'''''
Control Theory The Discrete LTI Dynamical System • The state solution to discrete LTI system
is
• The important point is that the natural response is dictated by matrix power Ak rather than the matrix exponential etA as before.
• The state dynamics are dictated by the matrix Hk = Ak • By diagonalizing we have Ak = TΛkT-1 • The dynamics are controlled by eigenvalue powers λk rather than etλ
1/27/14 USPAS 25
kk
kkk
CxyBuAxx
=
+=+1
∑−
=
−−+=1
0
10
k
ik
ikkk BuAxAx
Stability of LTI System Discrete Systems • Let λ be an eigenvalue of A
• Dynamics of eigenmode for λ are determined by value of λk
• Magnitude-angle representation of λ on the complex plane
• That is, ρ = |λ|, and θ =ang λ
• This yields
• θ = 0 indicates no oscillation
• |λ| = 1 indicates (marginally) stable oscillation
• |λ| < 1 indicates exponential decay
• |λ| > 1 indicates exponential growth
1/27/14 USPAS 26
Let λ = ρe jθ ρ,θ ∈ R
Unstable: exponentially
increasing
Non-oscillatory
Re λ
Im λ
+1 -1
-i
+i
Asymtotically stable: exponentially
decreasing λ k = ρ ke jkθ = ρ k coskθ + iρ k sinkθ
Looks like a Z transform
Stable, oscillatory
Control Theory Controllability and Observability of a Discrete LTI System • State solution to the discrete LTI system
• The system is controllable iff we can steer to any state η ∈ Rn • There exists a k > 0 and sequence {u0, u1, …, uk-1} such that yk = η .
• LTI controllability criteria: 1. Matrix (I+A+…+Am-1)B has full row rank 2. Augmented matrix [B | AB | … | Am-1B] has full row rank
• That is, there are n linearly independent columns
• Observability is analogous to the continuous case • Augmented matrix [CT | (CA)T | … | (CAk-1) T]T has full row rank
1/27/14 USPAS 27
xk =Akx0 + Ak−1−iBui
i=0
k−1
∑
Classical Control Example Stabilization: The Role of Feedback
• Typically in classical control we pick a feedback control law of the form
u(t) = Ky(t)
• The matrix K is chosen to move the system response (eigenvalues λ1 and λ2 of A) from the right half-plane to the left half-plane • Location in C determines feedback response • Open loop stability depends upon • Closed loop stability depends upon • Show this!
1/27/14 USPAS 28
Re λ
Im λ
Asymtotically stable: exponentially
decreasing
Unstable: exponentially
increasing
λ1
λ2
Note: “Classical Control” refers to the use of transfer functions in frequency domain to design feedback loops. Modern methods are more general, more applicable, and do not necessarily involve feedback
feedback
λ1
λ2
etA
et (A+KCB)
Classic Control Example: Disturbance Rejection • What if control signal u(t) is perturbed by a small amount δu(t)?
• That is u → u + δu
• Let us choose a special perturbation
• Then the (open loop) perturbed response is given by
1/27/14 USPAS 29
⎪⎩
⎪⎨
⎧
>+
<
= ∫ −1
)(
1
1 for)(
for)(
)(
1
ttdet
ttt
tt
t
t Bvx
x
x A ττ
t
δu(t) =0 ∈ Rm for t < t1v ∈ Rm for t > t1
"#$
%$
x(t)
x1(t)
x0
t1
Bv
Closed System
x(t) =Ax(t)+Bu(t) A ∈ Rm×m, B∈ Rm×n
y(t) =Cx(t) C∈ Rk×m
e(t) = y(t)− y0u(t) =Ke(t) K ∈ Rm×k
Classical Control Example Linear Regulator with Disturbance Rejection
1/27/14 USPAS 30
Σ
Controller K e error Dynamical
System (beamline)
environmental disturbance
δu Plant G
user output/
“readback”
u y plant input
plant output
(beam position)
Δ
user input/ “set-point”
y0
(desired beam position) Controller
• How do we get this to work??? • How fast to react? • Ignore certain frequencies? • etc.
Classical Control Example Design of Linear Regulator
• Recall governing equations for continuous LTI plant with linear feedback K
• Let’s try a scalar example • a = 1, b = 1 • c = ½,
• Choose • y0 = 2 • δu = 3
• Feedback • k = ??
1/27/14 USPAS 31
x(t) =Ax(t)+Bu(t) A ∈ Rm×m, B∈ Rm×n
y(t) =Cx(t) C∈ Rk×m
e(t) = y(t)− y0u(t) =Ke(t) K ∈ Rn×k
Pick k = -3/2 a+bkc = 1+1(-3/2)½ = ¼
Pick k = -3 a+bkc = 1+1(-3)½ = -½
Stability governed by matrix A + BKC
x(t) =Ax(t)+BKe(t) =Ax(t)+BK Cx(t)− y0( ) = (A+BKC)x(t)−BKy0
Classical Control – A Final Word The Transfer Function
• Transfer function H(s) of plant G is input/output relation in frequency domain • Take Laplace transform ∫ · e-stdt of everything in sight (s is on the right half plane)
• Stability of H(s) requires that s cannot be an eigenvalue of A • This will not happen if all eigenvalues are on the left half plane as before!
• Lets apply this to the perturbed regulator
• It can be shown this is equivalent to existence of for Re(s) > 0
1/27/14 USPAS 32
sx(s) =Ax(s)+Bu(s)y(s) =Cx(s)
⇒ y(s) =C A− sI( )−1Bu(s) ⇒ H(s) ≡C A− sI( )−1B
e =Hu− y0u = δu+Ke
have solution eu
!
"#
$
%&=
0 (I−HK)−1H(I−KH)−1K (I−KH)−1
!
"
##
$
%
&&y0δu
!
"##
$
%&&Loop equations
Thus, the regulator is stable if (I - KH)-1 exists and is stable for all s (or equivalently (I - KH)-1 )
I −K(s)−H(s) I
"
#$$
%
&''
−1
Control and LTI Systems Review
• Most LTI system can be put into state variable form • First-order, n-dimension matrix-vector ODE or difference equation
• For continuous case stability is determined by the matrix exponential etA
• For discrete case stability is determined by the matrix power Ak
• Stability ⇒ Where are the eigenvalues of A!?
• Closed loop stability depends upon A + BKC
1/27/14 USPAS 33
Supplemental Material • More detail on Linear System theory
1/27/14 USPAS 34
Accelerator Control Controlling a “Deadbeat System”
• Controlling accelerator systems is atypical • Individual shots cannot be controlled (speed of light) • Shots are correlated • Noise • Discrete systems in location index k • Continuous systems in time t • Sampled systems in time tk
• Many applications require “deadbeat controllers” • Feedback control that steers system to set-point in fewest number of steps
• Matching • Orbit correction • Twiss parameter observer
1/27/14 USPAS 35
Accelerator Control A Beam Position State Observer
1/27/14 USPAS 36
Accelerator Control Perturbations: Control and Orbit Difference • The former type of perturbation (of the control signal) is of the
type used for orbit difference applications • A magnet value along the beamline is perturbed from its nominal value. • The perturbed orbit remain identical to the nominal orbit until it reaches
the perturbed magnet, from there it diverges according to the effects of the magnet
• By subtracting the nominal trajectory from the perturbed trajectory we can observe the first-order response of the magnet
• We may perform the same procedure using a model of the beamline then compare the two magnet responses. Such a tool is valuable in diagnosing beamline irregularities.
1/27/14 USPAS 37
Linear Dynamical Systems Motivation • The material on stability and control is important for…
• Light sources, where beam positions must be maintained to tight tolerances
• RF systems where, for example, resonant tuning is essential in a highly disruptive environment
• We restrict the analysis to linear dynamical systems, however… Most beamlines are designed to be linear systems • At least most be treated as such • The XAL online model is designed using these principles.
1/27/14 USPAS 38
The State Representation Putting Linear Differential Equations in State Variable Form
Earlier we questioned the meaning of y = G(u) = L-1(u), well here it is…
• Start with our 2nd order linear differential equation
• Define our state variables x1 and x2
• Differentiate x2 yielding
• Arranging into matrix-vector format
1/27/14 USPAS 39
uybby
bbytutybtybtyb +−−==++
2
0
2
1012 or),()()()(
yxyx≡
≡
2
1
uxbbx
bbyx +−−== 1
2
02
2
12
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=⎟⎟⎠
⎞⎜⎜⎝
⎛
2
1
2
1
2
0
2
12
1
01
1010
xx
y
uxx
bb
bb
xx
dtd
)()()()()()(tttttt
DuCxyBuAxx
+=
+=This has form
Eq: Constant Coefficient ODE (cont.) • Thus, 2nd order equation
in standard form
has the state representation where
1/27/14 USPAS 40
)()()()( 012 tutybtybtyb =++ )()()()()()(tttttt
DuCxyBuAxx
+=
+=
( ) 0,01
10
,10
2
0
2
1
==
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
DC
BAbb
bb
• The state representation is generally easier to solve on the computer.
• There is a plethora of literature on the state representation properties.
The Matrix Exponential What is etA?
• Say square matrix A admits a diagonalization
• For example,
• Then etA has a very simple form
• The explicit form of etΛ is very easy to compute…
1/27/14 USPAS 41
,...),diag(),(,21
1
λλ=
⊂∈Λ=
×−
ΛTTTA
nnnGL
( ) ( )
1
133
22
31321
21
322
3221
−Λ
−
−−−Λ
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⋅+++=
+Λ⋅
+Λ+Λ+==−
TT
TΛΛΛIT
TTTTTTITTA
t
tt
e
ttt
tttee
1
1111
3001
1111
1221 −
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛=A
[ ] 11
32
3211
211
−− Λ=Λ
+⋅⋅
+++=
TTTT nn
a aaae …
Formulae:
Matrix Exponential Exponential of a Diagonal Matrix
• Let Λ be diagonal with entries {λi}, then
• The stability (behavior) of etΛ and, hence, etA = TetΛT-1 is completely determined by the eigenvalues of A according to etλ
• The matrix T determines coupling between these natural modes
1/27/14 USPAS 42
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
==
∑
∑
∑
∑∞
=
∞
=
∞
=
∞
=nt
t
t
in
n
i
i
n
i
i
n
i
i
u
it
e
ee
it
it
it
ite
λ
λ
λ
λ
λ
λ
2
1
0
20
10
0
!
!
!
!ΛΛ
Matrix Exponential Notes on the General Case
• Say A is not diagonalizable • A Jordon form always exists so that A = TΛT-1 where Λ is the Jordan
block diagonal matrix (Λ has 1’s to the right of the diagonal)
• Once again etA = TetΛT-1
• The exponential etΛ is not as easy to compute this time but the qualitative results are the same. • (Λ is triangular and we have terms like λketλ floating around) • The eigenvalues {λi} of A (the diagonal of Λ) determine the stability of the
system according to etλ • The matrix T determines the coupling between the natural modes of the system
1/27/14 USPAS 43
Matrix Exponential Notes on the General Case (continued)
• Singular value decomposition does not work for decomposing the matrix exponential
• Factor A = UDVT where D is the diagonal matrix of singular values and now U and V are both in SO(n) since A is square
• However, since VTU ≠ I in general, etA ≠ UetDVT • For example, (UDVT)2 = (UDVT) (UDVT) ≠ UD2VT
• However, Jordan decomposition always exist and allow for generalized eigenvalues, i.e., eigenvalues with value zero. • The natural modes for zero eigenvalues are called the center manifold of
the dynamical system
1/27/14 USPAS 44
The Matrix Exponential Existence
• Define exponential of a square matrix A by Taylor series
• Note for any A and t < ∞ that
• In fact for induced norm ||⋅|| and
1/27/14 USPAS 45
+⋅
+++=≡∑∞
=
33
22
0 322!AAAIAA ttt
nte
n
nn
t (well-defined operations)
0!
⎯⎯ →⎯∞→n
nn
nt A (well-defined values)
)(maxmax AA Λ=≤ λ
max
3max
32max
2
max
33
22
0
3221
322!
λ
λλλ
t
n
nn
t
e
ttt
tttnte
=
+⋅
+++≤
+⋅
+++≤= ∑∞
=
AAAIAA
Linear Systems: A Note on the Time-Varying Case
• The homogeneous solution is the generalization of the one-dimensional ODE
• If the coefficient a is a function of time, a = a(t), then the solution to the scalar ODE is
• This does not generalize! • The solution to is given by x(t) = Φ(t,0)x0 where Φ(t,0) is given by the Peano-Baker series
• (It is possible to define and say )
1/27/14 USPAS 46
0)( xx Atet =0)0();()( xxtaxtx ==
0)(
0)( xetxt
da∫=ττ
( ) ∫≡t
t daaL0
)( ττ ( ))()0,( tet tL AΦ =
0)0();()()( xxxAx == ttt
∫ ∫ ∫∫ ∫∫ ++++=t t tt tt
dtdtdttttdtdtttdttt0
10
20
33210
10
2210
11
1 21
)()()()()()()0,( AAAAAAIΦ
Transfer Function Approach Relationship of H(t) and Ĥ(s)
• Recall that in frequency domain x and u are related by
where Ĥ(jω) is the Laplace transform of etA evaluated at s = jω.
• If A is diagonalizable, i.e., A = TΛT-1, then
• Thus, Ĥ(s) has “poles” at s = {λi} = Λ(A) • By the Residue Theorem and Laplace Transform
• If the poles are in the right half plane Ĥ(jω) does not exist (is unstable) • If the poles are in the left half plane Ĥ(jω) exists for all ω (is stable)
1/27/14 USPAS 47
)(ˆ)(ˆ)(ˆ ωωω jjj uBHx =
( ) ( ) ( ) ( ) )(maxRefor)(ˆ 1
0
1
00
AAIAIH AIAIA Λ>−=−−=== −∞=
=
−−−∞
−−∞
− ∫∫ ssesdtedteest
tststtst
( ) TΛITH 11)(ˆ −− −= ss
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