Chapter 2: Structures - unipi.itpeople.unipi.it/.../256/2019/10/2_CSI_STRUCTURES.pdf · •...

Chapter 2: Structures

Motivation• Historical developments• System complexity• Model based analysis and limitations• Frequency shaping design for classical SISO

systems

Background• Frequency shaping design for classical SISO systems• Control loop components• Performance requirements in SISO systems

Review of Chapter 1

References:1. Classical control theory2. Bode sensitivity handout3. Skogestad – “MULTIVARIABLE FEEDBACK CONTROL Analysis and design”, chapters 1, 2, 5

(relevant parts)

Chapter 2: Structures

Basics of Systems Theory• State variable representation• Structural properties• MIMO properties and comparison with SISO

systems

MIMO Control Approaches• Pole placement• Linear Quadratic Optimal Control (LQR)• State Reconstruction and Estimation

Chapter 2: Systems Theory

• Systems theory is the interdisciplinary study of systems. A system is a cohesive conglomeration of interrelated and interdependent parts that is either natural or man‐made. Every system is delineated by its spatial and temporal boundaries, surrounded and influenced by its environment, described by its structure and purpose or nature and expressed in its functioning.

Definition …WIKI


• The goal of systems theory is systematically discovering a system's dynamics, constraints, conditions and elucidating principles (purpose, measure, methods, tools, etc.) that can be discerned and applied to systems at every level of nesting, and in every field for achieving optimized behavior.

.

.

.

.

.

.

1

m

u

u

1

p

y

y

System described by a mathematical model

• Multiple inputs and outputs• Interconnected dynamic variables• Finite dimensional, continuous• Linear and Nonlinear• Directionality

1

1

11 1 2

1 2

( , ,...)

...

( , ,...)p

p

n

n

n

rrn

d yf u u

dt

d yf u u

dt

ìïïï =ïïïïïíïïïïï =ïïïî

( ))

(??(

)y s

G su s

= =


• State variables representation is a convenient way (sometimes the only way) to describe multi‐input multi‐output nonlinear and linear dynamic systems.

( ) ( ( ), ( ))

( ) ( ( ), ( ))

t f t t

t g t t

ìï =ïíï =ïî

x x uy x u

( )

( )

( )

n

m

p

t

t

t

ìï Î Âïïï Î Âíïïï Î Âïî

x

u

y

• In the case of linear time invariant systems, we can take advantage of well established mathematical tools such as:

• Matrix theory• Linear algebra• Geometry, vector spaces• ...

( )

( )

( )

n

m

p

t

t

t


x

u

y

( ) ( ) ( )

( ) ( ) ( )

t A t B t

t C t D t

ìï = +ïíï = +ïî

x x uy x u


State Variables Derivation Reference material: Class Handouts (SYSTEM THEORY PRIMER 1, 2)Kailath: “Linear Systems”, Prentice Hall, 1980Dr. Dawson, Lecture notes, ECE 801, Clemson UniversitySkogestad, “Multivariable Feedback Control”, Chapter 4, relevant parts

• From the theory of linear and nonlinear dynamic systems, any set of differential equations can be transformed into an equivalent system of first order differential equations. In the case of LTI systems, we have:

( )

( )

( )

n

m

p

t

t

t


x

u

y

( ) ( ) ( )

( ) ( ) ( )

t A t B t

t C t D t


x x uy x u

1

1

11 1 2

1 2

( , ,...)

...

( , ,...)p

p

n

n

n

rrn

d yf u u

dt

d yf u u

dt

ìïïï =ïïïïïíïïïïï =ïïïî

• Definition: the vector x(t) is called state vector and it consists of the smallest number of independent variables necessary to uniquely describe the system (with known input) in the subspace .nÂ

• Note: For linear systems, the quadruple (A, B, C, D) is called realization. It is not unique and, given an input vector, an infinite number of state vectors belonging to the same subspace exists, which yields the same output.


cos

sin

cos co

s

s

in

m u vr wq X mg

m w uq v

m v wp ur

p

mg

Z m

Y

g

é ù- + = + Q

é ù- + = - Qê úë û

é ù- + = + Q Fê ú

ìïïïïí Fê úë ûïïïïî ë û

2 2

( ) ( )

( ) (

( )

)

( )xx zz yy xz

zz yy xx zx

yy xx zz zxI q I

I p I I qr r pq I L

I r I I pq rq p

I rp p r

I N

I M+ - + -

+ - - + =

+ - + - =

ìïïï=ïíïïïïî

• System linearized with respect to an equilibrium condition corresponding to rectilinear motion at constant velocity (blue equations above).

Rigid Body motion of a vessel

( , )f

u

v

w

p

q

r

=é ùê úê úê úê úê ú= ê úê úê úê úê úê úë û

x x u

x

• For the entire vehicle

• 6 State variables in speed (6 DOF); 12 state variables in position


0

0

0 0 0 0

0 0 ( ) 0

0 0 0

u u T

v r v v r RUD PROP

v z r v r r

m X u X u X X

m Y Y v Y m Y u Y v Y T

N I N r N Y u N r N

d

d

d

d

é ù é ù é ù é ù é ù é ù- -ê ú ê ú ê ú ê ú ê ú ê úê ú ê ú ê ú ê ú ê ú ê ú- - + - - - = +ê ú ê ú ê ú ê ú ê ú ê úê ú ê ú ê ú ê ú ê ú ê ú- - - - -ê ú ê ú ê ú ê ú ê ú ê úë û ë û ë û ë û ë û ë û

( ) xu u T PROP DIST

m X u X u X T t- - = +

sin

cos sin

( ) ( )zz yy xx zx

m u vr wq X mg

m v wp ur Y mg

I r I I pq rq p I N

ì é ùï - + = - Qï ê úë ûïï é ù- + = + Q Fí ê úë ûïï + - + - =ïïî

• Linearized Planar Maneuvering Model (surge, sway, yaw)

0

0

( )v r v v r

v z r v r r

RUD

m Y Y v Y m Y u Y v Y

N I N r N Y u N r N

r

d

d

d

d dy

é ù é ù é ù é ù é ù- - - - - -ê ú ê ú ê ú ê ú ê ú+ =ê ú ê ú ê ú ê ú ê ú- - - - - -ê ú ê ú ê ú ê ú ê úë û ë û ë û ë û ë û= -

=

• Linearized Planar Maneuvering Model (sway, yaw)


• Example: Consider a system governed by the following differential equations:

1 2 1 1 2

2 2 1 2

4 6 3

4 2

y y y u u

y y y u

+ - = -+ - =

1. Define as a state variable each output and its derivative to the order n – 12. Build the system of first order differential equations for the state variables3. Construct the output – state – input system

1 1

2 1

3 2

4 2

x y

x y

x y

x y

ìï =ïïï =ïï= íï =ïïï =ïïî

x

1 2

2 4 1 1 2 1 1

3 4 2 3

4 4 1 2

4 6 3,

4 2

x x

x x x u u y x

x x y x

x x x u

ìï =ïïï ìï= - + + - =ï ïï= =í íï ï= =ï ïîïï = - + +ïïî

x y

0 1 0 0 0 0

6 0 0 4 1 3

0 0 0 1 0 0

4 0 0 1 2 0

1 0 0 0

0 0 1 0y

é ù é ùê ú ê úê ú ê ú- -ê ú ê ú= +ê ú ê úê ú ê úê ú ê ú-ê ú ê úë û ë ûé ùê ú= ê úê úë û

x x u

x

• The D matrix is zero, which defines a strictly propersystem

• This is equivalent to a MIMO transfer function matrix with more poles than zeros


• Example

• Consider a system governed by the following differential equation:

- 6 8 2y y y u+ =

• Use analog diagram representation of the above:

1 2 12 1

2 1

8 6, 2( 8 6 )

x u x xy u x x

x x

ìï = - +ï= = - +íï =ïî

x

6 8 1

1 0 0

12 16 2

u

y u

é ù é ù-ê ú ê ú= +ê ú ê úê ú ê úë û ë ûé ù é ù= - + ê úê ú ë ûë û

x x

x

• The D matrix is non zero, which defines a proper system• This happens when the highest output and input derivatives are of the same

order• This is equivalent to a SISO transfer function with the same number of zeros as

poles


State Variables Solution

• The solution requires finding the time evolution of the state vector x(t), knowing its initial conditions and the input u(t). The output y(t) is then computed as a linear combination of state variables and inputs.

0( ) ( ) ( ),

( ) ( ) ( )n n n m

p n p m

t A t B t

t C t D t´ ´

´ ´

= += +

x x u xy x u

• Frequency Domain (Laplace transform)

0( ) ( ) ( )

( ) ( ) ( )

s A s B s

s C s D s

= += +

sx - x x uy x u

1 10

1 10

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

s sI A sI A B s

s C sI A C sI A B s D s

- -

- -

= - + -

= - + - +

x x u

y x u u

1

( ) ( ) ( )

( ) [ ( ) ]p m

s G s s

G s C sI A B D-´

=

= - +

y u

• The input – output relationship is characterizedby a pxm transfer function matrix

NOT DEFINED !( )

( )( )s

G ss

=yu


• Time Domain (from Theory of differential equations)

0

0

0

0

( ) ( )0

( ) ( )0

( ) ( )

( ) ( )

tA t t A t

tt

A t t A t

t

t e e B d

t Ce C e B d

t

t

t t

t t

- -

- -

ìïïï = +ïïïíïïï = +ïïïî

ò

ò

x x u

y x u0 0

( ) ( ) ( ) , ( )

( ) ( ) ( )

t A t B tt

t C t D t

ìï = +ï =íï = +ïî

x x ux x

y x u

• Definition : for LTI systems, we define the state transition matrix (or matrix exponential):

0( ) A t te -

• The transition matrix is the only element needed to find the solution of the system in the time domain

{ }1 1 ( )Ate L sI A- -= -0

00( , ) ( ) ( ) ( )( , )

t

t

t t tt B dt t tt= F + Fòx x u

• For a linear time varying (LTV) system, the solution is given by:


• Several methods to find the exponential transition matrix

{ }1 1 ( )Ate L sI A- -= -

3 1

1 3A

é ù-ê ú= ê ú-ê úë û

31

13s

sAsI

)4)(2(3

)4)(2(1

)4)(2(1

)4)(2(3

8631

13

21

sss

ss

sssss

sss

s

AsI

tttt

tttt

At

eeee

eeeee

4242

4242

21

21

21

21

21

21

21

21

• Example

• From the inverse of a matrix definition: 1 ( )( )

Adj AA

Det A- =


• Cayley – Hamilton theorem: Each square matrix satisfies its own characteristic equation

2 32 3

2! 3! !

kAt kt t t

e I At A A Ak

= + + + + + + ....iA A A A= ⋅ ⋅

• Recall infinite power series expansion (extension of the scalar case):

11 1 0

( ) ... 0n nn

A A a A a A a I--D = + + + + =

1 21 2 0

...n n nn n

A a A a A a I- -- -=- - - -

1 21 11 2 0 2 01

... ...n n nn n

n nn n

A AA a a A a Aa A a A a I- -- -

+ -- -é ù- - - -ê úë - -û= =- -

1 2 1

1 21 2 0

2 11 1 1 2 0 2

2 0

0

2...

... ...

...

n n n

n nn

nn n

n

n

n

n

nna a A a A a I a

a A

A AA

a A a Ia a A

A a Aa -

- --

- -- - - -

+-

-

+

-

é ùé ù- - - - - - -é ù- - - -ê

= =-

- -

-ê úê úë û

ú -û

û

ë

ë

12 1

0 1 2 10

( ) ( ) ( ) ( ) ( )n

At n in i

i

e t I t A t A t A A ta a a a a-

--

=

= + + + + = å

2 32 312! 3!

at t te at a a= + + + +


22 2 1

0 1 2 11 ( ) ( ) ( ) ( )

2!it n

i i i i i n i

te t t t t tll l l a a l a l a l -

-" = + + + = + + + +

• Use the Vandermonde matrix to solve for the coefficients i

tλ

tλ

tλ

tλ

nnnnn

n

n

n

ne

eee

λλλ

λλλλλλλλλ

3

2

1

1

2

1

0

12

13

233

12

222

11

211

α

ααα

1

111

Similarity Transformation

• Given square matrix A, there always exists a non singular matrix T such as A can betransforemed into a similar matrix = T‐1AT, so that A=TT‐1.

Matrix Exponential in Matlab


• Invariance properties• Trace• Rank• Eigenvalues• Determinant• Minimal polynomial• ...

0 0

( ) ( ) ( ) , ( )

( ) ( ) ( )

t A t B tt

t C t D t


x x ux x

y x u

1 1 1T T A T AT- - -= = = = S q x x q q

1

0 0

( ) ( ) ( ) , ( )

( ) ( ) ( )

t t T B tt

t CT t D t

-ìï =S +ïï =Síï = +ïïî

q q uq x

y q u

• Of particular interest is the choice of similarity transformation T as the modal matrix M(matrix of eigenvectors of A)

1

21

0 0

0 0

0 0n

M AM

ll

l

-

é ùê úê úê ú= L = ê úê úê úê úë û

1 0

0 n

t

t

t

e

e

e

l

l

L

é ùê úê ú= ê úê úê úë û

1

1 1

0

0 n

t

At t

t

e

e Me M M M

e

l

l

L - -

é ùê úê ú= = ê úê úê úë û


1P AP J- = 1 1At JtA PJP e Pe P- -= =

• In general, for A with repeated eigenvalues, the modal matrix is composed of eigenvectorsand generalized eigenvectors leading to a Jordan Form J

1 0 0 0 1 0 0

0 0 0 0 0 0 0

0 0 1 0 0 0 1

0 0 0 0 0 0 0

i

i

i i

i

i

J I I R

ll

l ll

l

é ù é ùê ú ê úê ú ê úê ú ê úê ú ê ú= = + = +ê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úë û ë û

( )i iI R t tJt Rte e e el l+= =

22 02! !

nRt nt t

e I Rt R Rn

= + + + + +

1000100

)!2(10)!1(!21

2

12

t

ntt

ntttn

n


• Examples

1 2

1 1 1 1 1, ,

0 2 2 0 1A M v vl

é ù é ù é ù- - é ùê ú ê ú ê ú= = = =ê úê ú ê ú ê úë û- - -ê ú ê ú ê úë û ë û ë û

1 1

2 2

1 0 0 1 1 0 1 1, ,

0 2 0 1 0 10 0

t tt At

t t

e ee e

e e

- -L

- -

é ù é ùé ù é ù é ù- ê ú ê úê ú ê ú ê úL = = =ê ú ê úê ú ê ú ê ú- - -ê ú ê úê ú ê ú ê úë û ë û ë ûë û ë û

2

20

t t tAt

t

e e ee

e

- - -

-


*1 1

1 1 1 1 1, ,

0 1 1 0 1A M v vl

é ù é ù é ù- - é ùê ú ê ú ê ú= = = =ê úê ú ê ú ê úë û- -ê ú ê ú ê úë û ë û ë û

1 1 1 0 0 1 1,

0 1 0 1 0 0 0 1Jt t

tJ e e-

é ù é ù é ù é ù- -ê ú ê ú ê ú ê ú= = + =ê ú ê ú ê ú ê ú- -ê ú ê ú ê ú ê úë û ë û ë û ë û 0

t tAt Jt

t

e tee e

e

- -

-

é ùê ú= = ê úê úë û

* Indicates generalized eigenvector (see handouts)


Additional Notes

2. Solution of a linear time varying system ( A, B, C, D are time varying)

( ) ( )Atg t Ce B D td= +

1. Impulse response of a LTI system

0

( ) ( ) ( )t

At Ag t Ce e B d D tt d t t d-= +ò

1( ) ( ) ( ) ( )AtL g t CL e B DL t C sI A B D G sd -é ùé ù é ù= + = - + =ê ú ê úê úë û ë ûë û

• Assume P(t) to be a fundamental matrix. This implies:

With some matrix P(t) such that


3. Then we can prove that:


Structural Properties

• By structural properties we indicate those characteristics that can be extracted from the mathematical description of a dynamic system independently of its physical nature and its constitutive equations.

Reference material: Class Handouts (SYSTEM THEORY PRIMER 1, 2)Kailath: “Linear Systems”, Prentice Hall, 1980Dr. Dawson, Lecture notes, ECE 801, Clemson UniversityDr. Choi, Lecture Notes, ME 851, Michigan State University

Stability

• Stability is an intrinsic property of a dynamic system (nonlinear, linear,…), which indicates the nature of its time evolution, when perturbed from an equilibrium condition (in general due to perturbations in the initial conditions).

• For a nonlinear system, the stability is formally referred with respect to an equilibrium condition.

• For a linear system, stability is a property of the system itself (why?)


Bottom‐line


Definitions

• Consider a nonlinear system:0

( , , ) ,

( , , )

t

t

ìï =ïíï =ïî

x f x ux

y g x u

• A movement xE(t) is of equilibrium if it satisfies:0

0 ( , , ) ,

( , , )E E E

EE E E

f t

g t

ìï = =ïíï =ïî

x x ux

y x u

• The algebraic system can have more than one solution

• So the stability properties are ‘local’


( ) ( ) ( )E

t t td= +x x x• They refer to the time evolution of lim ( )t

td¥

x where

Internal Stability definitions (do not depend on the input)


0 , A B= +x x u x

E

d= +x x x• Consider the perturbed motion with respect to initial conditions

( ) E E E

A B A Bd- = + + - - x x x x u x u Ad d=x x

1. Stability can be studied by looking at the autonomous system only.

2. It is sufficient to study the stability of the origin.

3. Stability of the origin and stability of the system are the same.

4. The knowledge of the system matrix A provides all necessary information aboutinternal stability.

Conclusions:

Stability properties for linear systems


DEFINITION: The origin is unstable otherwise

Theorem: A LTI system is asymptotically stable iff all eigenvalues of A have strictly negativereal part.

Theorem: A LTI system is stable iff the eigenvalues of A have negative or zero real part.The eigenvalues with zero real part must have equal algebraic and geometric multiplicity.

Theorem: A LTI system is unstable otherwise.


Lyapunov Stability: Lyapunov’s stability theory provides a general mathematical framework for the study of internal stability of linear and nonlinear systems

• Based on the total energy of a mechanical system perturbed from rest condition.

• V is undefined otherwise.

( )( ') 0, 0 ( ) 0 ( ) 0 : ' , 'V V Vd d= $ > < £ " - < ¹x x x x x x x x

• V is negative definite (semi‐definite) in x ’ if:

( )( ) 0, 0 ( ) 0 ( ) 0 : ,V V Vd d= $ > > ³ " - < ¹x' x x x x x' x x'

• V is positive definite (semi‐definite) in x ’ if:

1( ) : nV ⋅ Â Â• Definition – Lyapunov Function: Consider a scalar function

then:


• If the origin of the linearized system (x = 0) is asymptotically stable, then the equilibrium state xE is asymptotically stable,

• If the origin of the linearized system (x = 0) is unstable, then the equilibrium state xE is unstable,

• If the origin of the linearized system (x = 0) is stable, then nothing can be said about the stability of the equilibrium state xE .

( ) 0E E

f= =x x

First Theorem of Lyapunov:Consider a nonlinear autonomous system with equilibrium state xE :

0

fAd d d

=

¶= =

¶

x x

x x xx

Compute the linearized system:


( ) 0E E

f= =x x

Direct Method of Lyapunov (second theorem): – LetV(.) be a Lyapunov function, continuous and with continuous partial derivatives. Consider the equilibrium state of an autonomous system:

LetV(x) > 0 except for at mostV(xE) = 0 and compute its total time derivative

() ( )V

V f¶

⋅ =¶

xx

Note: if the Lyapunov function V(x, t) depends explicitly on the time we have:

( , ) ( )V V

V t ft

¶ ¶= +

¶ ¶x x

x

The equilibrium state is stable if 𝑉 0

The equilibrium state is asymptotically stable if 𝑉 0

The equilibrium state is unstable if 𝑉 0

𝑉 0, 𝑎𝑛𝑑 lim→

𝑉 𝑥 ∞The equilibrium state is globally asymptotically stable if


Notes on the Second theorem• Sufficient Conditions: what happens if there is no Lyapunov function?• How do we find a Lyapunov function?

• Example 1 22

2 1 2 1

0;

0( 1) E

x x

x x x x

ì é ùï =ïï ê ú=í ê úï = - - - ê úï ë ûïî

x

• Consider a quadratic Lyapunov function which is continuous, with continuous derivative, positive definite and equal to zero only at equilibrium.

2 21 2

( ) ;TV x x P P I= + = =x x x

2 2 221 2 2 11 2 1

( ) ( ) 2 2 2 ( 1)( 1)

xVV f x x x x

x x x

é ù¶ ê úé ù= = = - -ê úê úë û - - -¶ ê úë ûx x

x

• Compute its total time derivative:

11 ( ) 0x V" < >x The origin is therefore an unstable equilibrium point


1 22

2 1 2 1( 1)

x x

x x x x u

ìï =ïïíï = - +- -ïïî

• Perform the same analysis:

2 2 221 2 2 1 21 2 1

( ) ( ) 2 2 2 ( 1) 2( 1)

xVV f x x x x x u

x x x u

é ù¶ ê úé ù= = = - - +ê úê úë û - - +-¶ ê úë û

x xx

22 1

( ) 0 ( 1)V u x x< < - x

• The origin is now an asymptotically equilibrium point (an example of the so –called Lyapunov control)

• Can we stabilize the system around the equilibrium with an appropriate control input?


• Internal Stability and the Lyapunov Equation for LTI systems

( ) ( )

( )

(

(

)

)

B tt A t

t C t

ìï =ïíï =ïî

+x xy x

u Choose a quadratic Lyapunov function of the form:

( ) ( )T T T T T T TV P P A P PA A P PA= + = + = +x x x x x x x x x x x

TA P PA Q+ =- ( ) TV Q= -x x x

( ) 0 0TV P= > " ¹x x x x 0TP P= >

Stability requires to verify the sign of V

( ) 0 : 0T TV Q Q Q= - < " = > x x x

Theorem: A linear autonomous system is asymptotically stable if and only if, for every Qsymmetric and positive definite, there exists a matrix P, also symmetric and positive definite,such that:

0TA P PA Q+ + =


0TA P PA Q+ + =

Theorem: Given a matrix A, All its eigenvalues have strictly negative part if and only if thereexists a matrix P symmetric and positive definite and a matrix Q symmetric and positivedefinite, such that:

Corollary: If a solution P of the Lyapunov equation exists, it is unique.

Theorem: Let the following are equivalent:n nA ´Î Â

1. All eigenvalues of A have strictly negative real part

2. For every Q = QT > 0, there exists a unique solution P = PT > 0 of:

0TA P PA Q+ + =


• Example: Consider the following Lyapunov equation:

1 00; , 0, 0

0 2T T TA P PA Q A P P Q Q

é ù-ê ú+ + = = = > = >ê ú-ê úë û• Select a symmetric and positive matrix Q :

11 12 11211 22 12

21 12 22 22

0 00

0 0

q q qQ q q q

q q q q

é ù é ù>ê ú ê ú= - > ê ú ê ú= >ê ú ê úë û ë û

• Solve forP:

11 12 11 12 11

12 22 12 22 22

1 0 1 0 0

0 2 0 2 0

Tp p p p q

p p p p q

é ù é ù é ù é ù é ù- -ê ú ê ú ê ú ê ú ê ú+ =-ê ú ê ú ê ú ê ú ê ú- -ê ú ê ú ê ú ê ú ê úë û ë û ë û ë û ë û

1111

12

2222

020 , 0

04

T

qp

p P P

qp

ìïïï = >ïïïï = = >íïïïï = >ïïïî

Chapter 2: Systems Theory, Controllability

1. Can we transfer the state from x(t0) at some instant t = t1 to any point of its subspace?2. How can we compute the input u(t) such that we can transfer the state from some x(t0) to a

desired value x(t1) in a finite time?3. How can we find the minimum (most efficient) u(t) such that point 2 holds ? 4. Given the output y(t), can we reconstruct the entire state vector x(t)5. How can we find the minimum set of outputs, which allows the complete knowledge of the

state vector?

• The following properties are valid for dynamic systems in general (LTI, LTV, NL). We will limit ourselves to LTI systems of the form:

0 , , , ,n m p

A B

C

ìï = +ïï Î Â Î Â Î Âíï =ïïî

x x ux x u y

y x0

0

( ) ( )0

( ) ( ) ; ( )tA t t A t n

tt e e B d tt t t- -= + Î Âòx x u x


Theorem: for LTI continuous systems we have: XC = XR. For LTI discrete systems we have : XC ≤ XR

• Proof: see, Proof of theorem

Definition – Controllable: A system is controllable from time t0, if it is possible to find a finite input u(t) such that we can bring the state from some initial condition x(t0) ≠ 0 to the final state x(t1) = 0 in a finite time interval [t0, t1].

{ }1

0

( )1 0

0 ( ), ( ) : ( ) 0t

At A t

tt M e e B dtt t t t-" > $ < + =òu u x u

Definition – Reachable System: A system is reachable from time t0, if it is possible to find a finite input u(t) to bring the system from x(t0) = 0 to a final state x(t1) in a finitetime interval [t0, t1].

{ }1

0

( )1 1

0 ( ), ( ) : ( ) ( )t

A t

tt M e B d ttt t t t-" > $ < =òu u u x

• Examples1 0 2

, (0) 00 1 1

uæ ö æ ö÷ ÷ç ç÷ ÷ç ç= + =÷ ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è ø

x x x

22

:a

a

æ ö÷ç ÷çÎ Â = ÷ç ÷ç ÷çè øx x

0.5

1( )

20

( )t

t

t

x t e dt t-= ò1

( )1 2

0

( ) 2 2 ( )t

t

t

x t e d x tt t-= =ò

• The system is uncontrollable. For any input, we can only reachstates for which:

1 1 0 , (0) 0

0 1 1u

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= + =÷ ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è øx x x The system is completely controllable, since x1(t) and

x2(t) are linearly independent.

( )1 2 2

2

1 1( ) ( ) ( )

1 1 1

( ) ( )1

X s X s U ss s

X s U ss

ìïï = =ïï +ï +ïíïïï =ïï +ïî


Lecture 20 Course 6241 MIT Open courseware

What about this system?


https://eyes.nasa.gov/dsn/dsn.html

Definition – Controllability Matrix : Given a LTI system:

Define controllability matrix B, the nxnm matrix given by: 2 1( , , , , )nB AB A B A B-= B

, , n mA B= + Î Â Î Âx x u x u

Theorem: A LTI continuous system is (completely) controllable if and only if its controllability matrix has full rank:

( )2 1) ( , , , , )nrank( rank B AB A B A B n-= =B

1 11 1 1( ) ( )

1 0 0( ) (0) ( ) ( )

t tAt A t A tt e e B d e B dt tt t t t- -= + =ò òx x u u

• Proof: from CH theorem applied to the solution


Popov‐Belevitch‐Hautus (PBH) controllability Test: The pair (A, B) is controllable if and only if

• Summary: the following statements are equivalent:


• Example:

Rank [B] = 4

2 2 2 2

0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0

0 0 0 0

0 0 0 0 0

0 0 0 0 0

t

t

t

m l g m l g JF

M mgl lm

M mgl lm

m m m

m m

mm

é ù é ùê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê ú= +ê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úë ûë û

x x

Rank [B] = 4


• Example:


𝑥

0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 0

𝑥

00001

𝑢

𝑦 𝐶𝑥,

• Check the controllability of the system.• Check the stability of the system.• What is the response to initial conditions?• Note that the matrix A is a Jordan form.

Chapter 2: Systems Theory, Observability

Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. The observability and controllability of a system are mathematical duals.

0 , , ,n p

C

BAìï =ïï Î Â Î Âíï =ï

+

ïî

ux xx x y

y x

• If a system is not observable, the current values of some of its state variables cannot be determined through output sensors. This implies that their value is unknown to the controller (although they can be estimated by various means).

Definition: A LTI system is unobservable if:

0 0 ( ) 0, t 0, 0At nt Ce= = " ³ " ¹ Î Ây x x


• Example 2 1

0 2

1 0y

æ ö÷ç ÷ç= ÷ç ÷ç ÷çè øé ù= ê úë û

x x

x

2 2

1 102

2 20

1

( )( )

( ) 0

(t)= ( )

t t

t

x t e te xt

x t xe

y x t

é ùæ ö é ù÷ç ê ú ê ú÷ç= = ⋅÷ ê úç ê ú÷ç ÷ç ê úè ø ê úë ûë ûx

1

2

( ) ( )

( ) 2 ( ) 4 ( )

x t y t

x t y t y t

æ ö é ù÷ç ê ú÷ç =÷ç ê ú÷ -ç ÷çè ø ê úë û

• From the output we have the first state directly and we can compute the second state, thus the system is observable.

1 0

0 3

0 1y

æ ö÷ç ÷ç= ÷ç ÷-ç ÷çè øé ù= ê úë û

x x

x

1 10 33 2 20

2 20

( )( ) , ( ) ( )

( )

tt

t

x t e xt y t x t e x

x t e x-

-

æ öæ ö ÷÷ çç ÷÷ çç= = = =÷÷ çç ÷÷ çç ÷ ÷ç çè ø è øx

• Example

• The first state is not observable and also unstable!


• Example0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

1 0 0 0 0 ,

Bu

y

é ùê úê úê úê ú= +ê úê úê úê úê úë ûé ù= ê úë û

x x

x

Definition – Observability Matrix: Given a LTI system, we define the observability matrix Cas:

( ) ( )2 2 1

1

...TT T

T T T T n

n

C

CA

CA C A C A C A C

CA

-

-

é ùê úê úê ú é ùê ú ê ú= =ê ú ê úê ú ê úë ûê úê úê úë û

C

• How can we check the observability of thesystem?

( ) ( )?t ty x


Popov‐Belevitch‐Hautus (PBH) observability Test: The pair (A, C) is observable if and only if

Theorem: A LTI system is (completely) observable if and only if the observability matrix has maximum rank:

( )2 1) ( , , , , )n TRank( Rank C CA CA CA n-= =C

Apply CH theorem to show that, given the output to unobservable initial conditions

if the system is unobservable, the null space of In not empty

( ) 0 ,At rNO NO

t Ce= = Î Ây x x

0 0,... 1iNO

CA i n= " = -x


• Summary: the following statements are equivalent:


Some additional comments on structural properties

0( ) ( ) ( ),

( ) ( ) ( )n n n m

p n p m

t A t B t

t C t D t´ ´

´ ´

= += +

x x u xy x u 1

( ) ( ) ( )

( ) [ ( ) ]p m

s G s s

G s C sI A B D-´

=

= - +

y uUnique map

Infinite maps (Realizations)

Realizations

There are cases where the system is not controllable, but the uncontrollable state are stable and/or the system is not observable, but the unobservable states are stable. These weaker conditions are defined as:

• Detectable system if all unstable modes are observable• Stabilizable system if all the unstable modes are controllable


In general, the state vector may have states which are controllable, uncontrollable, observable, unobservable. This produces 4 vector subspaces, in Rn, whose dimensions depend on the rank of the controllability and observability matrices, and (obviously) their complements.

• Kalman Decomposition:ˆ ˆˆ ˆ( ) ( ) ( )ˆˆ( ) ( )

t A t B t

t C t

ìï = +ïïíï =ïïî

x x u

y x

11 12 13 141 1

2 22 24 22 4

3 33 34

4 44

ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆˆ 0 0ˆ ˆ ˆ ˆ ˆˆ , , , 0 0ˆ ˆˆ 00 0

ˆˆ 00 0 0

A A A A B

A A BA B C C C

A A

A

é ù é ùé ù ê ú ê úê ú ê ú ê úê ú ê ú ê úê ú é ù= = = =ê ú ê úê ú ê úë ûê ú ê úê ú ê ú ê úê ú ê ú ê úê úë û ë ûë û

xx

xxx

• Define a similarity transformation: 1A T AT-= CO NC O NOT X X X Xé ù= ê úë ûA


1 11 12 1 1

22 222

2 2

ˆ ˆ ˆˆ ˆ( ) ( ) ( )ˆ ˆˆ ( )ˆ ( ) 0

ˆ ˆ( ) ( )

t A A t Bt

x tt BA

t C t

ì é ùé ù é ùé ùïï ê úê ú ê úê úï = +ï ê úê ú ê úê úï ê úê ú ê úê úïï ë ûë û ë ûë ûíïïï =ïïïïî

x x

ux

y x

• Controllable subsystem

2 22 24 2 2

44 44

2 2 4 4

ˆ ˆ ˆˆ ˆ( ) ( ) ( )ˆ ˆ ( )ˆ 0( ) 0

ˆ ˆˆ ˆ( ) ( ) ( )

t A A t Bt

tt A

t C t C t

ì é ùé ù é ùé ùïï ê úê ú ê úê úï = +ï ê úê ú ê úê úï ê úê ú ê úê úïï ë û ë ûë û ë ûíïïï = +ïïïïî

x x

uxx

y x x

• Observable subsystem

2 22 2 2

2 2

ˆ ˆˆ ˆ( ) ( ) ( ) ˆ ˆ( ) ( )

t A t B t

t C t

ìï = +ïïíï =ïïî

x x u

y x• Controllable and Observable subsystem,

minimal realization

• Uncontrollable and Unobservable subsystem

3 33 3 34 4ˆ ˆˆ ˆ ˆ( ) ( ) ( )t A t A t= +x x x


Main result: The transfer matrix of a LTI system contains ONLY the states that are simultaneously controllable and observable

( ) ( ) ( )

( ) ( ) ( )

t A t B t

t C t D t


x x uy x u

x0 u

x

y1 ( ) ( )y

uG s C sI A B D-= - +

1 ( ) ( )xu

G s sI A B-= -

0

1 ( ) ( )yx

G s C sI A -= -0

1 ( ) ( ) ( )xx

G s sI A H s-= - =

𝑮𝒖𝒚 𝒔 𝑪 𝒔𝑰 𝑨 𝟏𝑩 𝑫 𝑪𝟐 𝒔𝑰 𝑨𝟐𝟐

𝟏𝑩𝟐


( ) ( ) ( )

( ) ( ) ( )

t A t B t

t C t D t


x x uy x u

External Stability (Input – Output Stability)

• BIBS Stability : A LTI or LTV system is BIBS stable (Bounded Input Bounded State) if:

0

0

0

, 0 0

( ) , ( ) ( , ) ( )t

t

t

t t t t t u d

e d

d t t t e

ìï" " > $ >ïïïïíï" < " > = F <ïïïïîòu x 1 ( ) ( )x

uG s sI A B-= -

BIBO Stability: A LTI or LTV system is BIBO stable (Bounded Input Bounded Output) if:

0

0

0

, 0 0

( ) , ( ) ( ) ( , ) ( )t

t

t

t t t t C t t u d

e d

d t t t e

ìï" " > $ >ïïïïíï" < " > = F <ïïïïîòu y 1 ( ) ( )y

uG s C sI A B D-= - +


• An internally stable LTI system is always externally stable

• An externally stable LTI system is not necessarily internally stable

• Internal stability depends on the eigenvalues of the A matrix

• External stability depends on the poles of the transfer function matrix

• Only If the system is simultaneously controllable and observable, the eigenvalues of A are the same as the poles of the transfer function matrix

• If the system is either uncontrollable or unobservable or both, the poles of the transfer function matrix are a subset of the eigenvalues of A

• Loss of controllability (observability) is due to zero‐pole (pole‐zero) cancellation in going from time domain to frequency domain representation


• Zero – Pole cancellation, Loss of controllability

1 ( ) ( )xu

G s sI A B-= -

• Pole ‐ Zero cancellation, Loss of observability

0

1 ( ) ( )yx

G s C sI A -= -

• Parallel connection should not contain common poles


What is the matter with zeros anyway?


• A zero is defined as input blocking frequency• Responsible for performance limitations• Responsible for loss of structural properties• In the case of MIMO systems the computation is not straightforward since

the input blocking property requires the cancellation of the input frequency as well its direction


zi is called system zero or invariant zero

• Zeros from State Space Realization (Minimal realization)

0 ( ) ( )

sI A BxP s P s

C Du y

é ùé ù é ù - -ê úê ú ê ú= = ê úê ú ê úê úê ú ê úë û ë û ë û


• MIMO zeros have directions (eigenvectors associated to the generalized eigenvalue problem, the vector sk above)

• Example: given the system G(s), show that s = ‐2 is a transmission zero

• The zeros are those values for which P(s) loses rank


• Zeros from Transfer Function Matrix (Minimal realization)


• The Smith‐McMillan form is used to determine the poles and zeros of the transfer matrix of a system with multiple inputs and/or outputs

• Step Procedure1. From transfer matrix to Smith form G(s)P(s)2. From Smith form to Smith McMillan form P(s)M(s)

are


• Note:𝛼 𝑠𝛽 𝑠

𝛾 𝑠𝑑 𝑠


1 21

( ) ( ) ( ) ( )... ( )r

i ri

p s s s s sb b b b=

= =

1 21

( ) ( ) ( ) ( )... ( )r

i ri

z s s s s sa a a a=

= =

G(s) and M(s) have the same normal rank r

• Example


( ) ( ) ( )

( ) ( ) ( )

t A t B t

t C t D t


x x uy x u

Who we Are

Note: a Hermitian matrix (or self‐adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose * A=A

Sylvester Equation


Operations on complex matrices

0( ) ( ) ( ),

( ) ( ) ( )n n n m

p n p m

t A t B t

y t C t D t´ ´

´ ´

= += +

x x u xx u

1

( ) ( ) ( )

( ) [ ( ) ]p m

s G j s

G j C j I A B D

w

w w -´

=

= - +

y u

Definition: A matrix G(s) is a real rational matrix if all its elements are real rational functions of s. A real rational matrix G(s) <‐> (A, B, C, D) is proper if det[D] ≠ 0 A real rational matrix G(s) has a real rational inverse G-1(s) if and only if

If G(s) is also proper, its inverse exists if and only if det[G(∞)] ≠ 0 A real rational matrix G(s) is stable if all its poles i are such that

det[ ( )] 0,G s s¹ " Î

0i

l -Î È

Lemma: A proper and stable matrix G(s) has a proper and stable inverse H(s) => G(s)H(s) = I, if: det[G(∞)] ≠ 0 det[G(∞)] ≠ 0 has no roots such that

0i

z +Ï È


Lemma: Let G(s) be a stable matrix with a stabilizable and detectable realization:

Then G(s) has a proper and stable inverse if and only if : det[D] ≠ 0 and A - BD-1C is a stable matrix.

( )A B

G sC D

é ùê ú= ê úê úë û

Chapter 2: MIMO Control Approaches

NOTE: the control methods reviewed in the following are applicable to both SISO and MIMO systems, but they have the most significant use when applied to multivariable systems in a state space format.

• Pole Placement with static full state feedback• Pole placement with dynamic output feedback• Linear Quadratic Regulator (i. e. linear optimal control)• Pole Placement with static output feedback

Pole Placement (Eigenvalue Assignment) with static full state feedback

0( ) ( ) ( ),

( ) ( ) ( )

t A t B t

t C t D t

= += +

x x u xy x u

K= - +u x r

0

( ) , (0)

A BK B

C

ìï = - +ï =íï =ïî

x x rx x

y x

( ) ( )( )0

0

( ) ( )( )0

0

( ) ( )

( ) ( )

tA BK t A BK t

tA BK t A BK t

t e e B d

t Ce C e B d

t

t

t t

t t

- - -

- - -

ìïïï = +ïïïíïïï = +ïïïî

ò

ò

x x r

y x r


Why are we interested in pole placement?

• To be able to specify the closed loop transient response

• To specify:• the speed of response • rise time, overshoot• To stabilize the system• the location of the dominant poles• …


Control problem: find an appropriate gain matrix K, such that the closed loop system has eigenvalues belonging to a specified spectrum {des}.

• {des} = roots of det ( I – A+BK )

Existence of Solution• Theorem (systems with single input): Given any nth order polynomial p(s)

there exists a unique matrix K (row matrix) such that p(s) is the characteristicpolynomial of the closed loop matrix obtained by state feedback of K.

• Theorem (multi input systems): given a multi input linear system (A,B) of order n, with m inputs, complete pole assignment is possible if and only if the system is controllable (valid for single input as well). That is:

1,.., 1,..,2 1[ , , ,..., ] ( ) ( .)

i n i nn

i iRank Rank B AB A B A B n A BK desl l

= =-= = - ºB

In that case, the number of closed loop gain matrices K ( pxm ) is infinite


Example:

( )

0 1 0

2 3 5

1 0

u

y

ì æ ö æ öï ÷ ÷ï ç ç÷ ÷ï ç ç= +÷ ÷ç çï ÷ ÷-ç çï ÷ ÷ç çè ø è øïíïïï =ïïïî

x x

x

1

2

5( ) ( )

3 2yu

G s C sI A Bs s

-= - =- +

• Find a full state feedback controller such that the closed loop poles are located in in s = ‐1 and s = ‐10. This means that characteristic polynomial is:

2( 1)( 10) 11 10s s s s+ + = + +

• The system is in controllable companion form (see notes) thus controllable:

2 22 1

(5 3) 2 5 11 10s k s k s s+ - + + = + + 2 2

1 1

5 3 11 14 5

2 5 10 8 5

k k

k k

ì ìï ï- = =ï ïí íï ï+ = =ï ïî î

1

2

8 5 14 5x

u rx

é ùé ù ê ú= - ⋅ +ê ú ê úë û ê úë û


Example:

2( )

( 1)s

G ss s

-=

-

• The system is unstable and non minimum phase. We wish to design a controller such that the closed loop system has a steady state error to a unit step equal to zero.

22 1 1 2

det( ) (2 1 ) 2( ) 0CL

I A k k k kl l l- = + - + - + =

2 0.76 0.48 [ 2.24,2]Ks s = -+ +

1 1 1

0 0 2

1 0

u A Bu

y = C

é ù é ù-ê ú ê ú= + = +ê ú ê úê ú ê úë û ë ûé ù= ê úë û

x x x

x x

( )

( )CL

CL

A BK Br A Br

A A BK

= - + = += -

x x x1,2

0.38 0.5776jl = -


1

2

2( ) ( )

0.76 0.48CL CL

sG s C sI A B

s s- -

= - =+ +

1( ) 4.1667SS CL

y t CA B-= - = -

• P(s) ≈ -1/4.1667


• In order to avoid the prefilter, we must feedback the error dynamics:

11 0 1 2

( ) 0 11 1 ( 1)

s sG s

s s s

-é ù é ù- - -é ù ê ú ê ú= =ê ú ê ú ê úë û - + -ê ú ê úë û ë û

1 0 1

1 0 1

0 1

A Bu u

y = C

é ù é ù-ê ú ê ú= + = +ê ú ê ú+ê ú ê úë û ë ûé ù= ê úë û

x x x

x x

2 0.24K é ù= - -ê úë û2det( ) 0.76 0.48 0

CLI Al l l- = + + =

CLA A BK= -

1 0.24 0.24

3 0.24 0.24

0 1

r

y

é ù é ù- - +ê ú ê ú= +ê ú ê ú+ + -ê ú ê úë û ë ûé ù= ê úë û

x x

x

1 1 1 1 2 2 2

1 2

2

2 0.24 0. 4

(

2

)u k x k x k xk r y k r

x x r

- -= - = - + -= - + -

1

2

0.24 0.48( ) ( )

0.76 0.48CL CL

sG s C sI A B

s s- - +

= - =+ +


1( ) 1SS CL

y t CA B-= - =


Static Output Feedback: The problem of output feedback is divided into static and dynamic:

K K m p=- = ´u y( ) ( ) ( )

( ) ( )

t A t B t

t C t

= +=

x x uy x

( )

A BKC B

C

ìï = - +ïíï =ïî

x x ry x

• Theorem (Shapiro, 1980): Give a controllable and observable nth‐ order MIMO system with m inputs and p outputs, we can assign with static output feedback at most r = max(m, p) closed loop eigenvalues. There is NO control over the remaining n – r eigenvalues (Asymptotic Stability not guaranteed by static Output Feedback)


Comment to the theorem: In practical design, exact pole assignment can not be achieved, nor is required. What we can do is to find an output feedback gain matrix, which minimizes the distancebetween the set of desired eigenvalues and the actual achievable set.

• Algorithm:

1. Order actual and desired eigenvalues starting from the fastest to the slowest,2. Define a performance index J as:

3. Set counter equal to 0,4. Input the system matrices A, B, C, the desired eigenvalues i, the weight qi, and set an

initial guess for the feedback gain matrix K0,5. Compute the eigenvalues li of the closed loop system A – BK0C,6. Order the two sets of eigenvalues as in step 1,7. Compute J(K),8. Find the optimal K* using some gradiental procedure9. Compute the eigenvalues of A – BK*C

2

1,

( )i

i n

J K d=

= å ( ) ( ) ( ) ( ){ }2 22 Re Re Im Imi i i i i i

d q m l m lé ù é ù= - + -ê ú ê úë û ë ûwith

NOTE: complete control of the transient response is not limited to pole placement, but requires appropriate selection of the amplitude as well.


Dynamic Output Feedback: The state vector is reconstructed (estimated) asymptotically by the design of an appropriate dynamic system called (Luenberger) observer.

( )0

, (0)

ˆ

F G Hì æ öï ÷ï ç ÷ï ç= + ÷çï ÷ç =÷çí è øïïï =ïî

uy

y

x xx x

x

lim( ) 0t¥

- =xx


= -xe x

• System and error dynamics:

0 , (0)

A B

C

ìï = +ï =íï =ïî

x x ux x

y x

( )

( ) ( )

F G H A B

F G HC A B

F F A HC G B

= - = + + - - == + + + - - == + - + + -

x u y x ux u x x u

x u

e x xee

• G = B• F = A - HC• The eigenvalues of F should have strictly negative real part

• In order for: lim( ) 0t¥

- =xx

( )0

( ) ( ) A HC tA HC t e -= - =e e e e

• H is called observer gain matrix


• The computation of the observer gain matrix H is therefore a pole assignment problem on (A - HC)T = (AT-CTHT ) , thus the solution requires the pair (A, C ) be observable, which is the same as controllability of (AT, CT ).

System Eigenvalues

( )A B H C= + + -x x xu y


A B= +x x u• System

K= -u r x• Control law

( )A B H C= + + -x x xu y• Luenberger Observer

( )A HC= -e e• Error dynamics

20 0

;

0

n

A BK BK B

A HC

C

æ é ùé ù é ù- -ç ê úê ú ê úç = = +ç ê úê ú ê úç -ç ê úê ú ê úë û ë ûë ûç Î Âççç é ùç =ç ê úë ûççè

x

q q r

q

y q

e

• The overall closed loop system is:

• The closed loop system matrix eigenvalues are the union of those of (A-BK) e (A-HC); this property is called separation principle, which shows that closed loop system dynamics and observer dynamics are decoupled.


Example: ( 1)( )

( 1)( 5)-

=+ +s

G ss s

.1.33 1.35

20

jìï- ïíï-ïî

0.7

1.9 rad / sec

0=

n

step

xwe

• Design Requirements• (need an integrator)• Closed loop system of

order 3

3

0 0

0 0

6 5 0 1 0

1 0 0 0 0

1 1 0 0 1

aug

aug

A Bu

x C r

u r

é ù æ ö æ ö æ ö÷ ÷ ÷ç ç çê ú ÷ ÷ ÷ç ç ç= + +÷ ÷ ÷ç ç çê ú ÷ ÷ ÷-ç ç ç÷ ÷ ÷ç ç çè ø è ø è øê úë ûæ ö æ ö æ ö- - ÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷- ÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø

x

x

x3

1 1

2 2

( )x r y

x

dt

x

x x

é ùé ù ê úê ú ê úê ú = ê úê ú ê úê ú ê úê úë êë û⋅ ú-û ò


( 1)( )

( 1)( 5)s

G ss s

-=

+ +

( 1)( )

( 1)( 5)AUG

s sG s

s s s-

=+ +

Controllable System

16.6800 96.5976 45.7120

u K= - =é ù= - ê úë û

x

x

• Full State feedback pole placement controller

Full State Feedback


• State Observer: The integrator added for the steady state requirement is an unobservable state, we need to add a measurement of that state.

1 1 0

0 0 1 aug

æ ö- ÷ç ÷ç= ÷ç ÷ç ÷çè øy x

• Added Sensor

• Observer synthesis:

-40, -50, -60• Choice of observer dynamics:

-154.2500 258.25 -1.0

0 0 40TH

æ ö- ÷ç ÷ç= ÷ç ÷ç ÷çè ø

• Observer Gain Matrix:

Unobservable System


6 5 0 1 0

1 0 0 0 0

1 1 0 0 1aug aug

u r

æ ö æ ö æ ö- - ÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç= + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷- ÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø

x x1 1 0

0 0 1 aug

æ ö- ÷ç ÷ç= ÷ç ÷ç ÷çè øy x

u Kx= -

( )A Bu H C= + + - yx x x

• Observer error • Closed Loop Step Response

Pole Placement with Matlab

Observer Design with Simulink


Linear Quadratic Regulator (Linear Optimal Control over infinite Horizon)

• To formalize an optimal control problem we need 3 main components:• The Model of the System;• The Constraints on state and control variables;• The expression of the Performance Index to be minimized/maximized.

• The Performance Index quantifies the performance of the controlled system.• It reflects State and Input Constraints• It is usually described by a scalar Functional, which relates a real number to each

admissible solution of the state vector, and control vector time histories.

• Optimal Control Methods express specifications directly in the time domain.

• They determine the control action as the one minimizing/maximizing an appropriate Performance Index (Cost Function J ).

* * *

,min ( , ) ( , ) ( , )x u

J x u J x u J x uÎ

é ù £ê úë û( )* * * *

,argmin ( , ) , : ( , ) ( , )

x uJ x u x u J x u J x u

Î

é ù = £ê úë û

• Optimal Control methods can produce either open loop or closed loop optimal input strategies


• The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory.

• The brachistocrone path is a minimum time path of a point mass to go from point A to point B under the influence of weight.

Optimal Control by ... wiki

Optimal Control via Calculus of Variations• Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in

functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equations of the calculus of variations.


• The above are necessary conditions only !• 2PBVP

Bolza Problem


Linear Quadratic Optimal Control (LQR):


0 0

T TT T T

A B A BH Q AQ Ax

¶¶

ìï = + é ùé ù é ù é ùïïï ê úê ú ê ú ê ú = +í ê úê ú ê ú ê úï - -= - = - - ê úê ú ê ú ê úï ë û ë ûë û ë ûïïî

x x u x xu

x lll l0 0

( )

( ) ( ) ( ) f f f

t

t S t t

ìï =ïíï =ïî

x xxl

• Euler – Lagrange equations:

* 10 0 ( ) TT THR B t R B

u¶¶

-= + = = -u u ll

• Optimal Control Law:

( ) ( ) ( )t S t t= xl

* 1( ) ( ) ( ) ( ) ( )Tt R B S t t K t t-= - = -u x x Feedback Control

( )1 0 , ( )T Tf f

S A S SA Q SBR B S S t S-+ + + - = = Matrix Differential Riccati Equation (DRE)


The previous problem is a typical Regulator control system. Tracking instead, is the design of a controller such that the closed loop system is capable of following some specified input (reference) signal.

Example: Tracking Regulator (Servomechanism)

( )

x Ax Bu

r t

ìï = +ïíï =ïî

desired trajectory

( ) ( ) ( ) ( )0

1 12 2

ft

T T Tf f f f f

t

J S Q R dté ù

= - - + - - +ê úê úë ûòx r x r x r x r u u

10 Tu

H R B-= = -u l

( )0

f f f f f f fS S

ìïïïíï = - = +ïïî

x

x r x al( ) ( ) ( )

f f ft S t t= -a r

( )1

T

T

HS S

R B S-

ìï æ öï ¶ ÷çï ÷= - = + +çï ÷ç ÷çí ¶è øïïï = - +ïî

x x ax

u x a

l

( ) ( )( )1 1 0T T T TS A S SA Q SBR B S SBR B A Q- -+ + + - + - - - = x a a r

( )1

1

T T

T T

S A S SA Q SBR B S

a SBR B A a Qr

-

-

ìï = - - - +ïïíï = - +ïïî

( )

( )f f

f f f

S t S

t S

ìï =ïíï = -ïîa r


1( ) ( ) ( ) ( )Tt R B S t t t- é ù= - +ê úë ûu x a• The optimal control law becomes:

0 1 0 1; ( )

2 1 1 0x x u r t

é ù é ù é ùê ú ê ú ê ú= + =ê ú ê ú ê ú-ê ú ê ú ê úë û ë û ë û

We wish x1 to follow a unit step and x2 to go to zero

• Example:

( ) ( )15

0

2 2 21 1

0

1 1 0.0025tf

ft

J x x u dt=

=

é ù= - + - +ê úê úë ûò• Build an appropriate Performance Index

( )1( ) ( ) 200 0 1 ( ) ( ) ( )Tu t R B t S t t a t- é ù= - = - +ê úë û xl

( ) ( ) ( )f f f

a t S t r t= -

( )( )

21 2 2

2 2 4 2 1 42

4 4 4 2

1 2 2

2 4 2 1

200 4 2

200 2

200 2 2

200 2 2

200 1

S S S

S S S S S S

S S S S

a S a

a S a a

ìï = - -ïïïï = + - -ïïï = + -íïïï = - +ïïï = + -ïïî

1

2

4

1

2

2

0

0

2

0

f

f

f

f

f

S

S

S

a

a

ìï =ïïï =ïïïï =íïï = -ïïïï =ïïî


0 min

0; 0

T T

uT T

J x Qx u Ru d

Q Q R R

t¥ é ù= +ê úë û

= ³ = >ò; ;n m px u yÎ Â Î Â Î Â

0 0

( ) ( ) ( ), ( )

( ) ( ) ( )

t A t B tt

t C t D t


x x ux x

y x u

Linear Quadratic Optimal Control (Infinite Horizon):

• Controllability is necessary to guarantee that J exists and it is finite

• It can be shown that if the system is controllable (stabilizable) the solution of the differential Riccati equation

( )1 0; ( ) 0T Tf

S A S SA Q SBR B S S t-+ + + - = =

Tends asymptotically to the constant positive (semi) definite constant solution S of the algebraic Riccati equation (ARE)

1 0 T TA S SA Q SBR B S-+ + - =

1( ) ( ) ( )Tt K t R B S t-= - = -u x x

• The optimal controller and the closed loop system are given by (regulator):

( )0

( )

( ) A BK t

A BK

t e -

ìï = -ïïíï =ïïî

x x

x x


Closed Loop System and Stability

• Assume the pair (A,D) be observable, where DTD=DDT=Q, then

all the state variables trajectories are represented in xTQx. 0 0

0 0T AtD e = =x x

Lyapunov Lemma: If (A,D) is observable, Q > 0, and A < 0, there exists a unique positive definite solution to the Lyapunov equation:

ATS+SA+DDT=0, with A = A-BK

• So closed loop stability is guaranteed from observability of (A,D) [or detectability (weaker condition)]

; T T T TC C C C C Q= = =y x y y x x

• NOTE: any observable linear system with independent measurements satisfies the requirements


Summary

1( ) ( ) - ( ) Tt R B S t K t-= - =u x x• is given by a full state feedback structure:

1 T

T T

A BR B

Q A

-é ù-ê úê ú- -ê úë

é ù é ùê ú ê ú=ê ú ê úê ú ê úëû û ûë

x x

ll

1. The controller gives a global minimum for J2. It is the only solution of the problem3. It stabilizes the closed loop system4. The associated Hamiltonian matrix has its 2n eigenvalues symmetric

w.r.t. the imaginary axis5. The Hamiltonian matrix does not have eigenvalues on the imaginary

axis and this implies that ARE has a solution and the solution is unique

0 0

( ) ( ) ( ), ( )

( ) ( ) ( )

t A t B tt

t C t D t


x x ux x

y x u

• Given a controllable and observable LTI system

0

0; 0

T T

T T T T

J Q R d

Q Q C C CC R R

t¥ é ù= +ê úë û

= = = ³ = >ò x x u u

• The optimal control law that minimizes

https://www.mathworks.com/help/control/ref/lqr.html


1 T

T T

A BR B

Q A

-é ù é ù é ùê ú ê ú ê ú= =ê ú ê ú ê ú

é ù-ê úê ú- -ê úê ú ê ú ê úë û ëë ûë û û

x x x

l llH

• Main Point: In the solution of the LQR (we limit ourselves to the infinite horizon case), the Hamiltonian matrix does not have eigenvalues on the imaginary axis and this implies that ARE has a solution and the solution is unique

1 0 T TA S SA Q SBR B S-+ + - =

• NOTE: The theory related to Riccati equations and Hamiltonian matrices can be found in Zhou, Doyle: “Robust and Optimal Control”, Chapter 13.

• We have studied the Lyapunov equation in Chapter 1 and have seen the roles it played in internal stability and structural properties of linear systems.

• A more general equation than the Lyapunov equation in control theory is the so‐called Algebraic Riccati Equation (ARE). Roughly speaking, Lyapunov equations are most useful in system analysis while AREs are most useful in control system synthesis; particularly, they play the central roles in H2 and H∞optimal control.

Some General Comments


• The relationship between Hamiltonian matrix and ARE solution is based on the choice of n –dimensional H invariant subspaces spanned by the 2n eigenvectors of H.

13.1

13.2

• Definition: an invariant subspace of a linear mapping T : V → V from some vector space V to itself. It is a subspace W of V that is preserved by T.

Jacopo Riccati


The relationship between solution X of ARE, Hamiltonian matrix H, and location of its eigenvalues is based on several theorems stated below:


• We are not interested in all the solutions of ARE, but those which are stabilizing, that is for which

( )iA RXl -+ Î

( )

( )

H dom Ric

X Ric H

ìï =ïíï =ïî


( )

( )

H dom Ric

X Ric H

ìï =ïíï =ïî

No eigenvalues of H on the imaginary axis

X form a basis of


How do all these theorems relate to the Linear Quadratic Regulator?

1 T

T T

A BR B

Q A

-é ù-ê ú= ê ú- -ê úë ûH

1 0 T TA S SA Q SBR B S-+ + - =

A RX+ 1 TA BR B S A BK-- = -

• Show that Lyapunov equation derived from a closed loop LQR is nothing but its Riccati Equation (except for a multiplying 2 term)


Im

Re

1 T

T T

A BR B

Q A


• 2n eigenvalues of the Hamiltonian Matrix

• n closed loop eigenvalues of LQR


Asymptotic Properties of LQR

1 T

T T

A BR B

Q A


• The closed loop eigenvalues (poles) of a LQR controller are the n stable eigenvalues of the 2nx2n Hamiltonian matrix

• The weighting matrices Q and R are the design parameters reflecting performance and control energy requirements. There are no theoretical results on their selection (except for selecting the elements as the inverse of the square of their maximal admissible value).

• Question: by changing the weights, is there a pattern for the closed loop poles?

det( ) 0sI - =H

1 T

T T

A BR B

Q A

-é ùé ù é ù é ù-ê úê ú ê ú ê ú= = ê úê ú ê ú ê ú- -ê úê ú ê ú ê úë û ë ûë û ë û

l llx x x

H

( ) ( ) ( )

( ) ( ) ( )

t A t B t

t C t D t


x x uy x u 0

,T T TJ Q R d Q C Ct¥ é ù= + =ê úë ûò x x u u

1( ) ( ) - ( ) Tt R B S t K t-= - =u x x 1 0 T TA S SA Q SBR B S-+ + - =


• Answer: we are looking for similarities with the classical Root Locus.

• Graphical equivalence is limited to single input single output systems. In this case:

0 T TJ dr t

¥ é ù= +ê úë ûò y y u u det( ) det 0

T

T T

BBsI A

sIC I A

rC s

é ùê ú-ê ú- = =ê ú

+ê úë û

H

• Define:

1

det( ) ( ) ( )

det( ) ( )

( )( ) ( )

( )

CL CL

OL

OL

OL

sI s s

sI A s

sG s C sI A B

s-

- = F F -- = F

Y= - =

F

H

• Use determinants algebra to compute the eigenvalues of the Hamiltonian matrix (see handouts)


1( 1) ( ) ( ) ( ) ( ) ( ) ( ) 0n

CL CL OL OL OL OLs s s s s s

r- F F - = F F - + Y Y - =

( ) ( )11 0

( ) ( )OL OL

OL OL

s

r

s

s s

Y Y -+ =

F F -• The closed loop poles are the stable poles of H and they change

with the parameter r

1 ( ) ( )( ) ( ) 0 ( ) ( ) 0

( ) ( )OL OL

OL OL OL OLOL OL

s ss s s s

s sr- Y Y -

F F - + = F F - =F F -


1 ( ) ( )( ) ( ) 0 ( ) ( ) 0

( ) ( )OL OL

OL OL OL OLOL OL

s ss s s s

s sr- Y Y -

F F - + = Y Y - =F F -

• Assume order of OL(s) OL(−s) is 2m < 2n


2

0 1 12 1 0 0( )1

1 2

usG s

sy

ì é ù é ùïï ê ú ê úï = +- ï ê ú ê úï= í ê ú ê úë û ë ûï- ï é ùï = ê úï ë ûïî

x x

x

• for k = 1

0T T TA P PA C C PBB P+ + - =

• Symmetric Root Locus (SRL)


2

1( )

5 6

sG s

s s

-=

+ +


2

1( )

( 6)( 4 6)s

sG s

s s--

+=

+


Stability Margins of LQR

• Question: Is there a relationship between the LQR controller and the relative stability?

1( ) ( )c

L s K sI A B-= -

Loop Transfer Function

1( ) ( )G s C sI A B-= -

System Transfer Function

• NOTE: In SISO systems the loop breaking point is irrelevant. In MIMO systems the loop transfer matrices change depending on the location of loop breaking point. In addition, matrix multiplication is not commutative!

• Outline of procedure:• Select LQR generated Gain Matrix Kc = R-1BTS,• Compute the return difference transfer matrix Euclidean Norm:

• |1+L(s)| or ||I+L(s)||• Use spectral factorization and the Lyapunov and Riccati equations

equivalence


• Answer: Computation of the return difference transfer function yields the so –called Kalman’s Inequality

1 11 ( ) 1 ( ) 1T

c cK sI A B K sI A B- -é ù é ù+ - - + - ³ê ú ê úë û ë û

21 ( ) 1L s+ ³*[1 ( )] [1 ( )] 1L s L s+ + ³ 1 ( ) 1L s+ ³

• The Kalman’s inequality holds for SISO as well as MIMO systems, as long as the weight on the controls is diagonal, that is R = rI.

2( ) 1I L s+ ³


• From Nyquist criterion:


Example: Homicidal Chaffeur as Differential Game

• The homicidal chauffeur problem is a mathematical pursuit – evasion problem which pits a hypothetical runner, who can only move slowly, but is highly maneuverable, against the driver of a motor vehicle, which is much faster but far less maneuverable, who is attempting to run him down. Both runner and driver are assumed to never tire. The question to be solved is: under what circumstances, and with what strategy, can the driver of the car guarantee that he can always catch the pedestrian, or the pedestrian guarantee that he can indefinitely elude the car?

• Original dynamic models were Dubin's car for the pursuer and a single integrator for the evader

• Objective: to reduce relative distance to less than some value

2 2( ) ( ) ( )

f f ft t t= -P Ez x x

0

0

, ( )

, ( )P P P P P

E E E E E

A B u t

A B v t


x x xx x x

0

2 2 21 1minmax ( ) ( ) ( )

2 2

f

P E

t

f f fR Rt

J t t t dtì üï ïï ïé ùï ï= + -ê úí ýï ïê úë ûï ïï ïî þ

òu vz u v


Similar Problem: bullfighter must follow specific movements to avoid a fast, charging bull


Example: Prisoner’s Dilemma as Discrete Game

the Nash equilibrium is a solution concept of a non‐cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his/her own strategy. If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

1. AL and BILL arrested independently for drug dealing and given each 2 years in jail2. AL and BILL suspected in participating to previous bank armed robbery3. District Attorney proposes AL the following:

• 1 year to AL if confesses robbery and 10 years to BILL if he denies• 10 years to AL if he denies and 1 year to BILL if he confesses• 3 years to AL if he confesses and 3 years to BILL if he confesses

4. District Attorney proposes BILL the same independently What is the best payoff for the two prisoners? What is the best payoff for the district attorney?

Global Optimum but UNSTABLE

NASH Equilibrium


Example: Suboptimal Closed Loop Control


Example: Suboptimal Closed Loop Control

Verify (AC)

Search (DC)

Assessment (AC)

Service (AC)


• Note 1: If the LQR controller does not have full state direct availability, since observability condition holds, we can use an observer.

• Note 2: The LQR can used used to easily compute parametric evaluations by changing the matrices Q and R.

• Note 3: Static output feedback can’t be used because of the known stability limitations

A B

C D


x x uy x u

, ,n m pÎÂ ÎÂ ÎÂ x u y K K m p=- = ´u y

• Theorem (Shapiro, 1980): Give a controllable and observable nth‐order MIMO system with m inputs and p outputs, we can assign with static output feedback at most r = max(m, p) closed loop eigenvalues. There is NO control over the remaining n – r eigenvalues (Asymptotic Stability not guaranteed by static Output Feedback)

Chapter 2: Optimal State Estimation (LQG)

• Question: Can we design a dynamic compensator, which is optimal (in a quadratic sense)?

• Answer 1: From a controller standpoint the answer is YES, since we could design a LQR (constant feedback gain matrix, guaranteed stability, good stability margins,...)

• Answer 2: From the state estimation standpoint the answer is NO, since we only have a Luenberger observer, which is guaranteed to be stable but not optimal


The answer to the second question is the development of an optimal state estimator (under some conditions) called Kalman Filter, and the use of the separation principle to connect it with an optimal LQR controller leading to the so‐called Linear Quadratic Gaussian (LQG) compensator.

Prof. Stengel's Notes (Princeton University)

• Although derived for a specific estimation problem, the Kalman Filter and its derivatives, is probably the most commonly used estimator in industry


Optimal state estimation in a quadratic sense (follower of Least Square Estimation)

0 0 0

( ) 0,

( ) 0,

( ) ,

t W

t V

t Q

ì é ùï ï ê úë ûïï é ùí ê úë ûïï é ùï ê úï ë ûî

w

v

x h

A B

C

ìï = + +ïíï = +ïî

x x u wy x v

• are symmetric constant matrices, called theintensity matrices of the two white noiseprocesses (indicating the variance). The initialstate is assumed to be a random vector.

0, 0W V³ >

• w(t), v(t) are wide sense stationary, white noise, uncorrelated, ergodic processes

0 0

ˆ ˆ ˆ( )

ˆ ˆ( )F

A B K C

t

= + + -=

x x u y xx x

{ }0 0, ,

min minF F

T

K x K xJ E Z= e eˆ

0Z

ìï = -ïíï >ïî

e x x

• We wish to find an estimator of the form:• Such that it minimizes the weighted

squared estimation error:

• Since the structure is fixed, the design entails finding the filter Gain Matrix KF

and the initial conditions:

0ˆ,

FK x


0 0 0 0

( )

ˆ( )F F

A K C K

t

ìï = - + -ïíï = - =ïî

e e w ve x x e

• Insert the Error Dynamics to modify the Performance Index to be minimized.

{ } ( )T Te e

J E Z Z tr Q t Zé ù= = + ê úë ûe e h h

• Minimize each term separately

{ }0 0ˆ 0E - =x x { } { }0 0

ˆE E=x x• Selection of the filter optimal initial conditions:

( ) ( )Q t Q t£ 1TFK QC V -=( ) ( )Q t Q tº • Selection of the filter optimal Gain Matrix:

• For a controllable system, and infinite horizon optimization

( ) 0( )Q t Q > const

1 0T TAQ QA W QC V CQ-+ + - = • KF Riccati equation (ARE)


LQG Control Synthesis

• Note that now all the signals and the performance index for the controller design are random processes and stochastic variables with Gaussian density functions.

( )0

( ) ( ) ( ) ( ) ( ) ( )ft

T T

t

J E Q R dt t t t t t t¥ì üï ïï ïï ï= +í ýï ïï ïï ïî þò x x u u

0, 0Q R³ >

• Find the optimal controller u(t) that minimizes: A B D

C

ìï = + +ïíï = +ïî

x x u wy x v

0 0 0

( ) 0,

( ) 0,

( ) ,

t W

t V

t Q

ì é ùï ï ê úë ûïï é ùí ê úë ûïï é ùï ê úï ë ûî

w

v

x h

0

ˆ ˆ ˆ ˆ( )ft

T T Tf f f

t

J E Q R dt S

¥ì üï ïï ïï ï= + +í ýï ïï ïï ïî þò x x u u x x

ˆ ˆ

ˆF

A B K

C

ìï = + +ïïíï = -ïïî

x x uy x

hh

ˆc

K=-u x K R B ScT 1 1 0T TA S SA Q SBR B S-+ + - =

1 0T T TAQ QA DWD QC V CQ-+ + - =1TFK QC V -=

Main Result:


• The performance of the closed loop system is computed in a stochastic sense, in terms of expected values and variances of the signals of interest.

0

0F F

F c F

A K C D K

K C A BK K

é ù é ù é ù- -ê ú ê ú ê ú= +ê ú ê ú ê ú-ê ú ê ú ê úë ûë û ë û

wq q

v

0 0 00

0 0ˆ

é ù é ù-ê ú ê ú= =ê ú ê úê ú ê úë û ë û

e xq

xh

hˆ

é ùê ú= ê úê úë û

eq

x C CA B= +q q x

{ } 11 12

21 22

( ) ( )x x

Txx x

Q QQ E E E

Q Q

é ùê úé ùé ù é ù= - - = ê úê ú ê úê úë û ë ûë û ê úë û

q q q q

• The statistics are given by the closed loop covariance matrix

0 0

00

0

( )

x x T TC C C C

x x

WA Q Q A B B

V

Q t Q

ì é ùïï ê úï + + =ï ê úí ê úë ûïïï =ïî

0F

CF c

A K CA

K C A BK



11 11

11 12 12

12 22 12 22

[ ] ( )

( ) ( )

( ) (0

0

)

0

x x T T TF F F F

x T T x T x TF C F F F

x T T T x T x x TF C F C F F

A K C Q Q A K C DWD K VK

Q C K Q A BK A K C Q K VK

Q C K Q A BK K CQ A BK Q K VK

ìï = - + - + +ïïï = + - + - -íïïï = + - + + - +ïî

Q t Qx11 0 0( )

Q t Q t12 0 22 0 0( ) ( )

11

22

0 0

00

x

x

Q

Q

é ù é ùê ú ê ú=ê ú ê úê ú ê úë ûë û

Filter Riccati SolutionLyapunov Equation Solution

12 0 12( ) 0 ( ) 0x xQ t Q t= =• Error and state estimate are uncorrelated since:

11( )xQ t• can be solved independently:


( ) 1( )

LQG C C F FK s K sI A BK K C K

-= - - + +

0

0ˆF F

F c F

A K C D K

K C A BK K

é ù é ù é ù é ù- -ê ú ê ú ê ú ê ú= = +ê ú ê ú ê ú ê ú-ê ú ê ú ê ú ê úë ûë û ë û ë û

e wq q

vx


DSN JPL

Example: Single Channel Rigid Antenna with noisy controller

0 1 0

0 1 0.1

1 0

x x u

y x

ì é ù é ùïï ê ú ê úï = +ï ê ú ê úï -í ê ú ê úë û ë ûïï é ùï = ê úï ë ûïî

{ }( ) 20 10 ( ) ( )u t t té ù= - +ê úë û x w

2

2 2

.0025 / 0( ) : [0, ]

0 0.05 / sec /w w

rad Hzt S S

rad Hz

é ùê ú= = ê úê úë û

w

• Compute the solution to the associated Lyapunov equation

0

0 1 0 0 .0025 0, ,

2 2 2 1 0 .05

T TAQ QA DWD

A D W

+ + =é ù é ù é ùê ú ê ú ê ú= = =ê ú ê ú ê ú- -ê ú ê ú ê úë û ë û ë û


• The Standard Deviation of the output is 0.087 rad = 5 degrees. This means that the controller is keeping the antenna position within 5 degrees from desired position for 67 % of the time.

{ }.0075 .015Q diag¥ =• The solution is found to be:

Chapter 2: Extras

2

2

2 2

2 3 2

12 3 2

2 3 2

1 13 2 2

s y u su sy y

y u su sy ysu u y y

ys ss s

y u y u ys s

= + - -

é ù= + - -ê úë û

= + - -

ì üï ïï ïé ù= - + -í ýê úë ûï ïï ïî þ

2 2 2 3s y sy y u su+ + = +

• Integral Operator: 1y y

s=

2 2 3y y y u u+ + = + 2

3 2( )

2 1

sG s

s s

+=

+ +

• Derivation from differential equations with derivative of the input

• Build an analog diagram using multiplication, integration and algrbraic addition

• Define as state variable the output of each integrator (the order is not important)

1 1 2

2 1

3 2

2

x u x x

x u x

= - += -

1

2

2 1 3

1 0 2

1 0

xu A Bu

x

y

é ù é ù é ù-ê ú ê ú ê ú= = + = +ê ú ê ú ê ú-ê ú ê ú ê úë û ë û ë ûé ù= ê úë û

x x x

x

1 2

2 2 1

2

3 2

x u x

x u x x

= -= - +

1

2

0 1 2

1 2 3

0 1

xu A Bu

x

y

é ù é ù é ù-ê ú ê ú ê ú= = + = +ê ú ê ú ê ú-ê ú ê ú ê úë û ë û ë ûé ù= ê úë û

x x x

x

2

3 2( )

2 1

sG s

s s

+=

+ +

Chapter 2: Extras

3 2 2( 7 14 8) (2 3 4)s s s y s s u+ + + = + +

2 2

3 2 3 2

(2 3 4) 2 3 4

( 7 14 8) 7 14 8

y s s w s w s w w

u s s s w s w s w s w w

= + + = ⋅ + ⋅ +

= + + + = ⋅ + ⋅ + ⋅ +

• Use the auxiliary w to verify the identity, rewrite input and output (u, y) as functions of w

2 33 2 22(2 3 4) ( 7 14 8)( 7 14 8) (2 3 4)s s s s s w ss s ss w+ + + = + ++ +++ +

• Differential operator: y sy=

• Draw the analog diagram using the highest derivative of w

( ) 7 ( ) 14 ( ) 8 ( ) 2 ( ) 3 ( ) 4 ( )y t y t y t y t u t u t u t+ + + = + +

Chapter 2: Extras

Kalman Filter Implementation: the discrete case

Chapter 2: Extras

Chapter 2: Extras

Simple Example 1

Simple Example 2

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