Control of PDEs - Introduction · Course Syllabus Control of PDEs Linear Systems General...

Course SyllabusControl of PDEsLinear Systems

Control of PDEsIntroduction

M. Grepl

Institut für Geometrie und Praktische Mathematik

RWTH Aachen

Wintersemester 2014/15

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General InformationCourse OutlineCourse Material

Date & Time

I LectureI Tuesday, 10.00-11.30am, Room 224.3I Start: 14.10.2014 (total 14 lectures)I Any conflicts?

I RecitationI Place and time to be determinedI Biweekly, 1.5 hours

I WebsiteI http://www.igpm.rwth-aachen.de/studium/OPT_PDE1

I Assessment (5 ECTS Credits)I Final exam (oral)I Date to be determined

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http://www.igpm.rwth-aachen.de/studium/OPT_PDE1



Instructor

Martin Grepl

I Room 126, Templergraben 55I Email: [email protected] Phone: 0241/80-96470I Office hours: by appointment

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Course Outline

I Linear and Nonlinear System Analysis (Review)I State-space FormulationI Lyapunov Analysis

I Backstepping ControlI Lyapunov StabilityI Parabolic PDEsI Observer DesignI Motion Planning

I Model Predictive Control (MPC)I Optimal Control and MPCI Dynamic Programming and MPCI Regulation & Stability

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Course Material

Primary SourceI Lecture notes

Reference TextsI M. Krstic, A. Smyshlyaev, Boundary Control of PDEs: A

Course on Backstepping Desgin, SIAM, 2008http://flyingv.ucsd.edu/krstic/b6.html

I J.B. Rawlings, D.Q. Mayne, Model Predictive Control: Theoryand Design, Nob Hill Publishing, 2009http://jbrwww.che.wisc.edu/home/jbraw/mpc/

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http://flyingv.ucsd.edu/krstic/b6.html

http://jbrwww.che.wisc.edu/home/jbraw/mpc/


MotivationSummary

Control of PDEs: Some Examples

6 / 48


MotivationSummary

Laser Welding

I Temperature governed by unsteady convection-diffusion equation

∂∂ty(t;µ) + v · ∇y(t;µ) = κ∇2y(t;µ) + q(x;µ)u(t)

I Equivalent volume heat source

q(x;µ) = e−x2

1/µ2(1)e−x2

2/µ2(2)e−x2

3/µ2(3)

Goal: Adaptive control (parameters µ = (µ(1), µ(2), µ(3))) toachieve desired weld pool depth

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MotivationSummary

Laser Welding

I Temperature governed by unsteady convection-diffusion equation

∂∂ty(t;µ) + v · ∇y(t;µ) = κ∇2y(t;µ) + q(x;µ)u(t)

I Equivalent volume heat source

q(x;µ) = e−x2

1/µ2(1)e−x2

2/µ2(2)e−x2

3/µ2(3)

Goal: Adaptive control (parameters µ = (µ(1), µ(2), µ(3))) toachieve desired weld pool depth

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MotivationSummary

Laser Welding – 2D Case

Temperature after start-up y(x, t = 0;µ) = 0

∂∂ty(t;µ) + Pe · ∂

∂xy(t;µ) = κ∇2y(t;µ) + q(x;µ)u(t),

whereI parameter µ = σ;I Laser velocity Pe;I u(t) control input (source strength).

Pe = vLc/!

!D

!

dW

x2

1

Measurement 1 Measurement 2

3.5 5x1

!N

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MotivationSummary

Laser Welding – 2D Results

Field variable: µ = 0.4, u(t) step input (N = 3720)

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MotivationSummary

Laser Welding – 2D Results

Parameter estimation & control: µ∗ = 0.46, sd,3(t) = 1

µIC = 0.463

εexp = 1%, fs = 5 Hz

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

u* (tk )

0 1 2 3 4 5 6 7 8 9 100

0.250.5

0.751

1.25

s 3(µ* ,tk )

0 1 2 3 4 5 6 7 8 9 1010

−410

−310

−210

−110

0

time t

|s3* −

s3(µ

* ,tk )|

µIC = 0.473

εexp = 5%, fs = 5 Hz

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

u* (tk )0 1 2 3 4 5 6 7 8 9 10

00.25

0.50.75

11.25

s 3(µ* ,tk )

0 1 2 3 4 5 6 7 8 9 1010

−410

−310

−210

−110

0

time t

|s3* −

s3(µ

* ,tk )|

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MotivationSummary

Radiation Therapy

Boltzmann transport equation: ψ ∈ Z × S satisfies

µ · ∇ψ(x, µ) + σt(x)ψ(x;µ)

= σs(x)

∫

Ss(µ · µ′)ψ(x, µ′)dµ′ + q(x)

whereI particle density distribution ψ;I cross section σt and scattering cross section σs;I scattering kernel s(·) (e.g. simplified Henvey-Greenstein)

s(y) =1− g2

4π(1 + g2 − 2gy)3/2

and parameter g depends on average cosine of scatteringangle;

I control input (source term) q(x);I Z ⊂ R3 and S is the unit sphere in R3.

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MotivationSummary

Radiation Therapy

Control problem:

minq(x)∈L2

ad(Z×S2)J(D) =

∫

Zi

(D − D)2dx+λ

2

∫

Z(q − q)2dx

where D(x) =

∫

Sψ(x, µ)dµ.

March 26, 2008 16:54 WSPC/103-M3AS 00278

Optimal Treatment Planning in Radiotherapy 589

α1/α2. On the other hand, the numerical effort increases as seen in the increasing

number of iterations. This corresponds well to similar observations for elliptic con-

trol problems and is due to the loss of uniqueness for the optimization problem in

the limit case α2 = 0.

5.3. Distributed control in 2D

The setup in our second 2D example was used as an approximation of a cross-

section of a human head1 to demonstrate the advantage of a transport calculation

over the diffusion approximation when the geometry contains a void-like region.

The slightly modified setup is shown in Fig. 1. Consider a 100 × 100 mm square.

The so-called cerebrospinal fluid (CSF) is represented by a thin layer around an

interior square. Contained in this square there are the tumour ZT and the spinal

chord ZR. The material parameters are summarized in Table 2.

Furthermore, we set

ψ =

1/4π in ZT

0 otherwise, α1 =

1 in ZR

25 in ZT, α2 = 1, Q(x, y) = 0

1 otherwise.

, (5.3)

Fig. 1. Geometry of the computational domain.

Table 2. Parameters of validation problem.

Tissue σa (mm−1) σs (mm−1)

CSF 0.001 0.01Other 0.05 0.5

Source: M3AS 18(4), 2008, 573-592 13 / 48


MotivationSummary

Radiation Therapy

March 26, 2008 16:54 WSPC/103-M3AS 00278

590 M. Frank, M. Herty & M. Schafer

(a) Dose in the optimal state. (b) Dose in the optimal state (clipping).

(c) Optimal control Q∗. (d) Optimal control Q∗ (clipping).

(e) Relative difference D−Dmax D

(f) Relative difference D−Dmax D

(clipping)

Fig. 2. Optimal state and optimal control.

The optimality system with isotropic scattering is solved by a QSP1 approximation

on a grid with approximately 10,000 elements.

In Fig. 2 we show three different aspects of the solution: dose distribution in the

optimal state, the optimal control and a relative difference plot between total doses

of target state and optimal state. The left-hand side shows the complete computa-

tional domain, whereas on the right-hand side one can find a zoom into the tumor

region. The thick black and white lines represent the geometry. The black circle

Source: M3AS 18(4), 2008, 573-59214 / 48


MotivationSummary

Hyperthermia Treatment

Bio-heat transfer equation

ρtct∂T

∂t= div(k∇T )−Wρbρtcb(T − Ta) +Q

where Q = σ/2|E|2 and

I ρt, ρb density of tissue and blood;I ct, cb specific heat of tissue and blood;I T, Ta temperature of tissue and arterial blood;I k thermal conductivity of tissue;I W blood perfusion;I Q power deposition within tissue;I σ electric conductivity; andI E electrical field (from Maxwell’s equations).

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MotivationSummary

Flow Control

Goal: Control position of shock in supersonic diffuser

Steady-state Mach contours

Figure 1: Steady-state Mach contours inside diffuser. Freestream Mach num-ber is 2.2.

1 Supersonic Inlet Flow Example

1.1 Overview and motivation

This example considers unsteady flow through a supersonic diffuser as shownin Figure 1. The diffuser operates at a nominal Mach number of 2.2, howeverit is subject to perturbations in the incoming flow, which may be due (forexample) to atmospheric variations. In nominal operation, there is a strongshock downstream of the diffuser throat, as can be seen from the Mach con-tours plotted in Figure 1. Incoming disturbances can cause the shock to moveforward towards the throat. When the shock sits at the throat, the inlet isunstable, since any disturbance that moves the shock slightly upstream willcause it to move forward rapidly, leading to unstart of the inlet. This is ex-tremely undesirable, since unstart results in a large loss of thrust. In order toprevent unstart from occurring, one option is to actively control the positionof the shock. This control may be effected through flow bleeding upstreamof the diffuser throat. In order to derive effective active control strategies, itis imperative to have low-order models which accurately capture the relevantdynamics.

1.2 Active flow control setup

Figure 2 presents the schematic of the actuation mechanism. Incoming flowwith possible disturbances enters the inlet and is sensed using pressure sen-sors. The controller then adjusts the bleed upstream of the throat in orderto control the position of the shock and to prevent it from moving upstream.In simulations, it is difficult to automatically determine the shock location.The average Mach number at the diffuser throat provides an appropriatesurrogate that can be easily computed.

1

Active flow control problem setup

Figure 2: Supersonic diffuser active flow control problem setup.

There are several transfer functions of interest in this problem. The shockposition will be controlled by monitoring the average Mach number at thediffuser throat. The reduced-order model must capture the dynamics of thisoutput in response to two inputs: the incoming flow disturbance and thebleed actuation. In addition, total pressure measurements at the diffuserwall are used for sensing. The response of this output to the two inputs mustalso be captured.

1.3 CFD formulation

The unsteady, two-dimensional flow of an inviscid, compressible fluid is gov-erned by the Euler equations. The usual statements of mass, momentum,and energy can be written in integral form as

∂

∂t

∫∫ρ dV +

∮ρQ · dA = 0 (1)

∂

∂t

∫∫ρQ dV +

∮ρQ (Q · dA) +

∮p dA = 0 (2)

∂

∂t

∫∫ρE dV +

∮ρH (Q · dA) +

∮p Q · dA = 0, (3)

where ρ, Q, H, E, and p denote density, flow velocity, total enthalpy, energy,and pressure, respectively.

The CFD formulation for this problem uses a finite volume method andis described fully in Lassaux [1]. The unknown flow quantities used are thedensity, streamwise velocity component, normal velocity component, and en-thalpy at each point in the computational grid. Note that the local flow

2

Oberwolfach Benchmark Collection16 / 48


MotivationSummary

Flow Control

Euler equations∂tρ+ ∂i(ρui) = 0

∂t(ρuj) + ∂i(ρuiuj) + ∂jp = 0

∂tE + ∂i((E + p)ui) = 0

whereI ρ fluid mass densityI ui, i = 1, 2, 3, fluid velocity vectorI E = ρe+ 1/2ρuiui total energy per volumeI e internal energy per unit massI p pressure

Measurements: pressure sensors — Control: bleed actuation

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MotivationSummary

Cooling of Steel Profiles

Temperate distribution

ρ cp∂T (x, t)

∂t= λ∆T (x, t),

T (x, 0) = T0,

λ(~n · ∇T ) = ui,

where ui controls the heat flux onbounday Γi.Control required sinceI Cooling process strongly influences

material propertiesI Large temperature gradients have to

be avoided (deformation, . . . )

Oberw. Bench. Coll.18 / 48


MotivationSummary

Contaminant Transport

I Application: control of emission (Source: Dede (2008))

I Application: Identification of sources

Airborne contaminants Airborne contaminantsin urban canyon. in LA basis.

Source: Bashir et. al. 2008 Source: Akcelik et. al. 2006

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MotivationSummary

Contaminant Transport

Dispersion of a pollutant (y(x, 0) = 0) Ω = [0, 4]× [0, 1]

∂∂ty(x, t) + U · ∇y(x, t) = κ∇2y(x, t) + gPS(x)u(t),

whereI source location at (xs1, x

s2);

I diffusivity κ;I U from Natural Convection (Navier-Stokes) flow;I u(t) control input (source strength).

x2

x1

ΩM8 ΩM

2(xs1, x

s2)

ΩM1κ

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MotivationSummary

Contaminant Transport – Sample Solutions

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MotivationSummary

Contaminant Transport – Sample Solutions

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MotivationSummary

And many more . . .

I Food processingI Process control (chemical reactions)I Reaction-Diffusion processesI Induction HeatingI Flow controlI . . .

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MotivationSummary

Remarks

I Control of PDEs also referred to asI control of infinite-dimensional systemsI control of distributed parameter systems

as opposed to finite-dimensional systems (Rn).

I Classical controller design for finite-dimensional systems based oninput/output description

I Transfer function are rational functions

I Direct controller design possible if transfer function is availableI Extension of classical design approaches (Nyquist, Passivity, . . . )I Infinite-dimensional controller must be approximated by

finite-dimensional system: late lumping

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MotivationSummary

Remarks

I Closed-form expression of transfer function for PDEs often notavailable

I 1. Step: finite-dimensional approximation of PDEI Indirect controller design (early lumping)

I Controller designed for finite-dimensional approximation does notnecessarily stabilize the infinite-dimensional system (spillovereffect)

I Further ProblemsI Finite-dimensional approximations are still high-dimensional

(Ricatti equation, Kalman Filter, Optimal Control)I Full state usually not available for feedback (boundary control or

observer design)I Possibly unknown parameters (adaptive or robust control)

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MotivationSummary

Methodologies

I Classical controller design

I Linear-Quadratic Regulators

I H∞ Control

I Backstepping Control

I Optimal Control (⇒ Optimierung C)

I Model Predictive Control (MPC)

I . . .

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State Space RepresenationSome Background

Control of Linear Finite-dimensional Systems

Reference (e.g.): T. Kailath, Linear Systems, Prentice-Hall, 1980

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State space representation

Linear Time-Invariant (LTI) System:

x(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t)

with initial condition x(0) = x0, whereI x(·) ∈ Rn is the state vector;I y(·) ∈ Rq is the output vector;I u(·) ∈ Rp is the control (or input) vector;I A ∈ Rn×n is the state (or system) matrix;I B ∈ Rn×p is the control (or input) matrix;I C ∈ Rq×n is the output matrix;I D ∈ Rq×p is the feedthrough matrix (often D = 0).

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State space representation

I Linear Time-Varying (LTV) System:

x(t) = A(t)x(t) +B(t)u(t)y(t) = C(t)x(t) +D(t)u(t)

I Discrete-Time-Invariant System:

xk+1 = Axk +Bukyk = Cxk +Duk

I Discrete-Time-Variant System:

xk+1 = A(k)xk +B(k)ukyk = C(k)xk +D(k)uk

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State space representation of PDEs

I Consider the one-dimensional heat equationwt = wxx, x ∈ (0, 1), t ∈ (0,∞)

w(0, t) = 0, t ∈ (0,∞)

w(1, t) = 0, t ∈ (0,∞)

w(x, 0) = ϕ(x), x ∈ [0, 1]

I Discretization: partition [0, 1] in equi-spaced finite grid, i.e.,xi = i∆x, i = 0, 1, . . . , N, where ∆x = 1

N

I FD semi-discrete equation for w = [w(x1) . . . w(xN−1)]T

wt =1

∆x2

−2 11 −2 1

. . . . . . . . .1 −2 1

1 −2

w = Aw

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State space representation of PDEs

I Finite element discretization (linear FE space)

∆x

6

4 11 4 1

. . . . . . . . .1 4 1

1 4

︸︷︷︸M

wt =1

∆x

−2 11 −2 1

. . . . . . . . .1 −2 1

1 −2

︸︷︷︸A

w

where M and A are the mass and stiffness matrix, respectively.

I Descriptor state space (DSS) model

Ex(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t)

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LTI Solution

Given the LTI system (or realization) A,B,C

x(t) = Ax(t) +Bu(t), x(0) = x0,y(t) = Cx(t), t > 0,

the solution x(t) is given by

x(t) = eAtx0 +

∫ t

0eA(t−τ)B u(τ ) dτ.

Note:I The matrix exponential is defined as

eAt =

∞∑

n=0

Antn

n!= I +At+

A2

2!t2 +

A3

3!t3 + . . .;

I The state-transition matrix is given by Φ(t) = eAt ;I Φ(t) takes the initial state x0 to a state x(t) by time t.

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DTI Solution

Given the DTI system A,B,C

xk+1 = Axk +Buk, x(0) = x0,yk = Cxk, k = 0, 1, . . . ,

the solution xk is given by

xk = Akx0 +

k−1∑

j=0

Ak−j−1B u(j)

Note:I The state-transition matrix is given by Φ(k) = Ak;

I Convolution integral turns into convolution sum.

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Relation LTI – DTI Solution

Given the LTI systemx(t) = Ax(t) +Bu(t), x(0) = x0,y(t) = Cx(t), t > 0,

with A nonsingular and given u(t).Assumptions:

I Sampling time ∆t, tk = k∆t;I Control u(t) piecewise constant.

The LTI solution at tk is then equivalent to the solution of the DTIsystem

xk+1 = Adxk +Bduk, x(0) = x0,yk = Cdxk, k = 0, 1, . . . ,

whereAd = eA∆t;Bd =

(eA∆t − I

)A−1B;

Cd = C. 34 / 48



Properties

Characteristic PolynomialThe characteristic polynomial of A is defined by

pA(λ) = det(λI −A)

Caylay-Hamilton TheoremEvery square matrix A satisfies its characteristic polynomial, i.e., if

pA(λ) = λn + a1λn−1 + . . .+ an−1λ+ an

is the characteristic polynomial of A ∈ Rn×n, then

An + a1An−1 + . . .+ an−1A+ anI = 0.

Note: Matrix exponential can be computed by a finite sum.

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Properties

Impulse response

The impulse response h(·) of a single-input single-output (SISO)realization A, b, c is the solution of

x(t) = Ax(t) + bδ(t), x(0) = 0,

where δ(t) is the Kronecker delta function. It thus follows that

h(t) = eAtb.

Duhamel’s Principle

Given the impulse response h(t), the response to an arbitrarycontrol input u(t) is given by

x(t) = h(t) ∗ u(t) =

∫ t

0h(t− τ )u(τ ) dτ.

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Notions of Stability

External Stability

A realization A, b, c is externally stable, if a bounded input

|u(t)| < M1, −∞ < −T ≤ t <∞,

produces a bounded output

|y(t)| < M2, −T ≤ t <∞.

I Necessary and sufficient condition: the impulse response satisfies∫ ∞

0|h(t)| dt < M <∞

I DTI System: the impulse response of the DTI system satisfies∞∑

0

|h(k)| < M <∞

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Stability

Internal (Asymptotic) Stability

A realization A, b, c is internally stable (or stable in the sense ofLyapunov) if the solution of

x(t) = Ax(t), x(t0) = x0, t ≥ t0tends toward zero as t→∞ for arbitrary x0.

I A system is stable if and only if

Re[λi(A)] < 0, i = 1, . . . , n,

where λi(A) are the eigenvalues of A.I DTI system: |λi(A)| < 1, i = 1, . . . , n.

I Internal stability implies external stability, but converse is not trueI External stability is equivalent to internal stability if realization is

minimal, i.e., both controllable and observable.38 / 48



Controllability

DefinitionA realization A,B,C is controllable, if any state x can bemoved to (or controlled to) any other state x by a suitable choiceof input (control) in a finite time interval.

I A realization A,B,C is (state) controllable if and only if then× np controllability matrix

C =[B AB A2B . . . An−1B

]

has full rank n.I Hautus Lemma for controllability: A realization A,B,C is

(state) controllable if and only if

rank[λI −A B

]= n, for all λ ∈ eig(A).

I Output controllability:

rank[CB CAB CA2B . . . CAn−1B

]= q

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Observability

DefinitionA realization A,B,C is (state) observable if we can uniquelydetermine the state x(t) given knowledge of A,B,C andy(t), u(t), t ≥ 0.

I A realization A,B,C is state observable if and only if thenq × n observability matrix

O′ = [C′ A′C′ . . . (A′)n−1C′]

has full rank n.I Hautus Lemma for observability: A realization A,B,C is

observable if and only if

rank

[λI −AC

]= n, for all λ ∈ eig(A).

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Duality of Controllability and Observability

Consider the realization A,B,C primal system

x(t) = Ax(t) +Bu(t), x(0) = x0,y(t) = Cx(t),

and the realization A′, C′, B′ dual system

xd(t) = A′xd(t) + C′ud(t), xd(0) = xd,0,yd(t) = B′xd(t),

Duality Principle

The primal realization A,B,C is controllable (observable) ifand only if the dual realization A′, C′, B′ is observable(controllable).

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State Feedback

Problem Statement: Given the LTI system

x(t) = Ax(t) +Bu(t), x(0) = x0,

find a nonsingular p× p matrix G and a p× n feedback gainmatrix K such that under state-feedback

u(t) = Gv(t)−Kx(t)

the new state equation

x(t) = (A−BK)x(t)+BGv(t), x(0) = x0,

can be assigned an arbitrary monic polynomial of degree n as thecharacteristic polynomial.

NoteState feedback can provide arbitrary relocation of the eigenvalues ofa realization A,B,C if and only if the system is controllable.

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State Feedback

I Feedforward gain G can be chosen to achieve a zero steady-statetracking error. Assume

I feedback gain matrix K is knownI number of controls is equal to number of outputs, i.e., p = q,

then y = v for x(t) = 0 if

G = −(C (A−BK)−1B

)−1

I Output feedback: u(t) = −Ky(t)

I Design MethodsI Controller canonical formI Direct methods (Bass-Gura)I . . .

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Observer Design

Given the realization A,B,Cx(t) = Ax(t) +Bu(t), x(0) = x0,y(t) = Cx(t), t > 0,

we define the (asymptotic) observer˙x(t) = Ax(t) +Bu(t) + L (y(t)− Cx(t))︸︷︷︸

error signalwith x(0) = x0 (initial guess) and L a feedback gain vector.

The error x(t) = x(t)− x(t) obeys the differential equation˙x(t) = x(t)− ˙x(t)

= (A− LC)x(t), x(0) = x0 − x0.

NoteIf the realization A,B,C is observable, the poles of A− LCcan be placed arbitrarily in the complex plane.

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Separation Principle

Consider the realization A,B,C with feedback controlu(t) = −Kx(t) and observer x(t).

We then obtain the closed loop system[x(t)˙x(t)

]=

[A −BKLC A− LC −BK

] [x(t)x(t)

],

[x(0)x(0)

]=

[x0

x0

]

with output y(t) = Cx(t) and characteristic polynomial

p(λ) = det(λI −A+BK) det(λI −A+ LC)

= pcont(λ) pobs(λ)

Separation PrincipleI Controller and observer can be designed separatelyI Overall system is stable if observer and closed loop system

(without observer) are stable45 / 48



A few more definitions . . .

I A realization A,B,C is minimal if and only if it iscontrollable and observable.

I A realization A,B,C is detectable if all unstable modes areobservable, i.e., all unobservable modes are stable.

I A realization A,B,C is stabilizable if all unstable modes arecontrollable, i.e., all uncontrollable modes are stable.Example: Consider the system

x(t) = Λx(t) +Bu, Λ = diagλ1, . . . , λn, λi 6= λj .

I The system is controllable, if and only if none of the elements ofB is zero

I The system is stabilizable, if Re(λi) < 0 holds for a zeroelement of B, i.e., Bi = 0.

46 / 48



Nonlinear Systems — next time . . .

Consider the nonlinear systemx1 = f1(t, x1, . . . , xn, u1, . . . , up)x2 = f2(t, x1, . . . , xn, u1, . . . , up)...

...xn = fn(t, x1, . . . , xn, u1, . . . , up)

with n states and p input variable. In vector notationx = f(t, x, u)

with q-dimensional output y given byy = h(t, x, u)

Special casesI Unforced state equation: x = f(t, x)I Autonomous (time invariant) state equation: x = f(x)

Next time: stability analysis (Lyapunov)47 / 48



Nonlinear Control Design Tools

ProblemThere is no single method that works for all problems

I Gain SchedulingI Feedback LinearizationI Sliding Mode ControlI Lyapunov RedesignI BacksteppingI Passivity-Based ControlI High-Gain Observers

References (for example):I J.-J.E. Slotine, W. Li. Applied Nonlinear Control, Prentice

Hall, 1991I H.K. Khalil. Nonlinear Systems (3rd edition), Prentice Hall,

200248 / 48

Control of PDEs - Introduction · Course Syllabus Control of PDEs Linear Systems General...

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