ece504: lecture 8 - worcester polytechnic institutespinlab.wpi.edu/courses/ece504_2008/8.pdf ·...

ECE504: Lecture 8

ECE504: Lecture 8

D. Richard Brown III

Worcester Polytechnic Institute

28-Oct-2008

Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 1 / 30

ECE504: Lecture 8

Lecture 8 Major Topics

We are still in Part II of ECE504: Quantitative and qualitative

analysis of systems

mathematical description → results about behavior of system

We will cover the following qualitative properties of systems:◮ Stability◮ Reachability and controllability◮ Observability◮ Minimality

Today, our focus is on analyzing the internal stability of LTI systems.We will also discuss techniques for analyzing the stability of nonlinearand time-varying systems.

You should be reading Chen Chapter 5 now. Sections 5.1 and 5.3-5.5all discuss internal stability.


ECE504: Lecture 8

Stability: Some Intuition

Suppose you have a CT-LTI system described by

x(t) =

−1 0 00 1 00 0 1

x(t) +

100

u(t)

y(t) =[

1 1 −1]

x(t) +[

0]

u(t)

Is this system “stable”?

Since this is an LTI system, we can compute the transfer function usingour standard technique:

g(s) = C(sI3 − A)−1B + D =(s − 1)2

(s + 1)(s − 1)2=

1

s + 1

Is this system stable?


ECE504: Lecture 8

Internal Versus External Stability

There are two types of stability that we can discuss when we usestate-space descriptions of dynamic systems:

1. Internal stability (are the states blowing up?)

2. External stability (is the output blowing up?)

Transfer functions can only tell you about the external stability ofsystems. State-space descriptions can tell you about both internal andexternal stability. This is another big advantage of the state-spacemathematical description with respect to transfer functions.

As we’ve seen, it is possible for systems to have external stability butnot internal stability.


ECE504: Lecture 8

Bounded Vector-Valued Functions

Definition

The vector-valued function z(t) : R 7→ Rn is bounded if there exists some

finite 0 ≤ M < ∞ such that ‖z(t)‖ ≤ M for all t ∈ R.

Recall that

‖z(t)‖ :=√

z⊤(t)z(t)

=√

z2

1(t) + · · · + z2

n(t)

is the Euclidean norm.


ECE504: Lecture 8

Internal Stability of Continuous-Time Systems

Definition

A continuous-time system is stable if and only if, when the input u(t) ≡ 0for all t ≥ t0, the state x(t) is bounded for all t ≥ t0 for any initial statex(t0) ∈ R

n.

Definition

A continuous-time system is asymptotically stable if and only if it isstable and

limt→∞

‖x(t)‖ = 0

for any initial state x(t0) ∈ Rn.

Note: Our “stable” is Chen’s “marginally stable”.


ECE504: Lecture 8

Continuous-Time Nonlinear System Example

Suppose we had a system with scalar state dynamics

x(t) = x(t)(1 − x2(t))

1. What are the equilibria of this differential equation?

2. Is this system (internally) stable?

3. Is this system (internally) asymptotically stable?


ECE504: Lecture 8

Internal Stability of CT-LTI Systems

Continuous time LTI system:

x(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

Note that internal stability has nothing to do with the output equationy(t) = Cx(t) + Du(t) or the matrix B. Internal stability is all about theproperties of the differential equation

x(t) = Ax(t) (1)

What are the equilibria of (1)?

Note that it is impossible to have multiple isolated equilibria when thesystem is LTI.


ECE504: Lecture 8

Remarks on Internal Stability

◮ If you are looking at a linear system, we can focus our attentionon the equilibrium point x(t) = 0. We don’t need to considerother equilibria in the nullspace of A because their behavior willbe the same as the equilibrium point x(t) = 0.

◮ If you are looking at a nonlinear system, the behavior at eachequilibrium point may be different. Each must be analyzedindividually.

◮ Our definition of asymptotic (internal) stability is appropriate forlinear systems because of the implicit equilibrium at the origin.

◮ Asymptotic (internal) stability may be defined differently fornonlinear systems.


ECE504: Lecture 8

Stability Criterion

Theorem

A continuous-time LTI system is (internally) stable if and only if both ofthe following conditions are true.

1. Re(λj) ≤ 0 for all j ∈ {1, . . . , s} where {λ1, . . . , λs} is the set ofdistinct eigenvalues of A.

2. For each λj that is an eigenvalue of A such that Re(λj) = 0, thealgebraic multiplicity of λj is equal to the geometric multiplicity ofλj, i.e. rj = mj.

Note that condition 2 is equivalent to A being “diagonalizable for alleigenvalues with zero real part”. Chen Theorem 5.4 requires eacheigenvalue with zero real part to have algebraic multiplicity one. It shouldbe clear that Chen’s condition is actually too strong.

Examples...Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 10 / 30

ECE504: Lecture 8

Stability Criterion Proof Sketch


ECE504: Lecture 8

Asymptotic Stability Criterion

Theorem

A continuous-time LTI system is asymptotically (internally) stable if andonly if Re(λj) < 0 for all j ∈ {1, . . . , s} where {λ1, . . . , λs} is the set ofdistinct eigenvalues of A.

Remarks:

1. It is easier to check asymptotic stability than stability.

2. Special name for matrices with all eigenvalues satisfying Re(λj) < 0:“Hurwitz” (the same Hurwitz as the Routh-Hurwitz stability criterionyou may have seen in an undergraduate systems/controls course).

3. The proof of this theorem follows directly from the fact that

limt→0

tmeαt = 0

for any integer m if Re(α) < 0.


ECE504: Lecture 8

Internal Stability of Discrete-Time Systems

The definitions of internal stability and asymptotic internal stability forcontinuous-time systems can be easily extended to discrete-time systems:

Definition

A discrete-time system is stable if and only if, when the input u[k] ≡ 0 forall k ≥ k0, the state x[k] is bounded for all k ≥ k0 for any initial statex[k0] ∈ R

n.

Definition

A discrete-time system is asymptotically stable if and only if it is stableand

limk→∞

‖x[k]‖ = 0

for any initial state x[k0] ∈ Rn.


ECE504: Lecture 8

Stability Criterion

Theorem

A discrete-time LTI system is (internally) stable if and only if both of thefollowing conditions are true.

1. |λj | ≤ 1 for all j ∈ {1, . . . , s} where {λ1, . . . , λs} is the set of distincteigenvalues of A.

2. For each λj that is an eigenvalue of A such that |λj | = 1, thealgebraic multiplicity of λj is equal to the geometric multiplicity ofλj, i.e. rj = mj.

The proof of this theorem is similar to the proof for the continuous-timecase except that the terms of Ak look like λk

j , kλkj , k2λk

j , etc.

◮ If |λj | < 1, then kmλkj is always bounded.

◮ If |λj | = 1, then condition 2 implies that there are only λkj terms, no

kmλkj terms.


ECE504: Lecture 8

Discrete-Time Asymptotic Stability Criterion

Theorem

A discrete-time LTI system is asymptotically (internally) stable if and onlyif |λj | < 1 for all j ∈ {1, . . . , s} where {λ1, . . . , λs} is the set of distincteigenvalues of A.

Remarks:

1. Special name for matrices with all eigenvalues satisfying |λj | < 1:“Schur” (the same Schur as the Schur-Cohn stability criterion).

2. The proof of this theorem follows directly from the fact that

limk→0

kmλkj = 0

for any integer m if |λj | < 1.


ECE504: Lecture 8

Discrete-Time Internal Stability Examples


ECE504: Lecture 8

Lyapunov Stability

◮ Lyapunov stability is a general internal stability concept that isparticularly useful for nonlinear differential/difference equations.

◮ The ideas of Lyapunov stability can also apply to LTI systems.

The main idea:

1. Assume the input u(t) ≡ 0 (we are looking at internal stability).

2. Let v(x(t)) be the squared “distance” of the state x(t) from theequilibrium of interest.

3. If v(x(t)) < 0 for all x(t) not at the equilibrium of interest, then thestate x(t) must be getting closer to the equilibrium. Why?

4. If the “distance” to the equilibrium is strictly decreasing for all x(t),then the system is asymptotically stable.

Recall that nonlinear systems may have many different equilibria but, inLTI systems, we only need to consider the equilibrium x(t) = 0.


ECE504: Lecture 8

Lyapunov Stability

t

v(x(t))

Note that v(x(t)) → 0 (the system is asymptotically stable) even thoughthis example shows a case when the “distance” to the equilibrium is not astrictly decreasing function of time. Hence, v(x(t)) < 0 for all x(t) impliesasymptotic stability, but the converse is not always true.


ECE504: Lecture 8

An Example of Lyapunov Stability for Nonlinear Systems

Let’s look again at the continuous-time nonlinear scalar differentialequation

x(t) = x(t)(1 − x2(t))

We know of three equilibria: x(t) ∈ {−1, 0, 1}. Let’s look at the Lyapunovstability of each of these equilibria...


ECE504: Lecture 8

An Example of Lyapunov Stability for Nonlinear Systems

x(t) = x(t)(1 − x2(t)) and v(x(t)) = (x(t) − ǫ)2 for ǫ ∈ {−1, 0, 1}

−1.5 −1 −0.5 0 0.5 1 1.5−10

−8

−6

−4

−2

0

2

x(t)

(d/d

t)v(

x(t)

)

ε = −1ε = 0ε = 1

None of the equilibria meet the requirements for Lyapunov stability. Why?Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 20 / 30

ECE504: Lecture 8

Positive Definite and Positive Semi-Definite Matrices

Definition

A matrix P ∈ Rn×n is positive definite if P = P⊤ and P has all

positive eigenvalues.

Remarks:

1. When P = P⊤, the matrix P is said to be symmetric.

2. Fact: All of the eigenvalues of real-valued symmetric matrices are real.

3. Note: Chen’s definition of positive definiteness does not requiresymmetry (his theorems say “symmetric and positive definite”).

Definition

A matrix P ∈ Rn×n is positive semi-definite if P = P⊤ and P has all

non-negative eigenvalues.

Obviously, any positive definite P is also positive semi-definite, but theconverse is not always true.


ECE504: Lecture 8

Quadratic Form

In linear algebra, we call expressions like z⊤P z the “quadratic form”.

◮ For any P ∈ Rn×n, z ∈ R

n, and any integer n ≥ 1, the result of thequadratic form is just a scalar.

◮ If P ∈ Rn×n is positive definite, then it is not too difficult to show

that z⊤P z > 0 for all z ∈ Rn and z 6= 0.

◮ Similarly, if P ∈ Rn×n is positive semi-definite, then we can show

that z⊤P z ≥ 0 for all z ∈ Rn.

Relationship to Euclidean norm:

◮ Recall that ‖z‖2 := z2

1+ · · · + z2

n = z⊤z = z⊤Inz.

◮ Is In positive definite?

◮ For positive definite P , we can define a generalized norm as‖z‖2

P:= z⊤Pz. Note that, like the Euclidean norm, this generalized

norm satisfies all of the intuitive properties of a distance measure.


ECE504: Lecture 8

Lyapunov Matrix Equation: CT-LTI Systems

Let P ∈ Rn×n be a constant positive definite matrix and define

v(x(t)) := x⊤(t)P x(t)

Assuming a continuous-time LTI system, we can write

v(x(t)) =

(

d

dtx⊤(t)

)

P x(t) + x⊤(t)P

(

d

dtx(t)

)

=

(

d

dtx(t)

)⊤

P x(t) + x⊤(t)P

(

d

dtx(t)

)

= (Ax(t))⊤ P x(t) + x⊤(t)P (Ax(t))

= x⊤(t)A⊤P x(t) + x⊤(t)P Ax(t)

= x⊤(t)(

A⊤P + P A)

x(t)

= −x⊤(t)Qx(t)

Is Q := −(

A⊤P + P A)

is symmetric? When must v(x(t)) < 0?


ECE504: Lecture 8

Lyapunov Theorem: CT-LTI Systems

Recall that, for CT-LTI systems:

A is Hurwitz ⇔ the CT-LTI system is asymptotically stable

Theorem

A is Hurwitz if and only if, for any positive definite Q ∈ Rn×n, there

exists a unique positive definite P ∈ Rn×n satisfying A⊤P + P A = −Q.

The proof of this theorem can be found in the Chen textbook Section 5.4.

The implied procedure:

1. You are given A ∈ Rn×n.

2. You pick a particular Q ∈ Rn×n that must be positive definite. A

good choice is In.

3. You then solve for P ∈ Rn×n. This must be done element by element

for each term in P .Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 24 / 30

ECE504: Lecture 8

Lyapunov Theorem: CT-LTI Systems

Example:

A =

[

−1 −10 −1

]

Clearly, this A is Hurwitz. Let’s try out the Lyapunov theorem...


ECE504: Lecture 8

Comments on the Lyapunov Theorem for CT-LTI Systems

◮ In general, solving for P in the equation A⊤P + P A = −Q meansthat there are n2 simultaneous linear equations to solve.

◮ The Lyapunov theorem has limited usefulness for low-dimension,e.g. n = 2 or n = 3, CT-LTI systems because it is usually easier tojust compute the eigenvalues from the characteristic polynomial.

◮ For larger dimensional problems, e.g. n ≥ 5, it might be easier towork with the Lyapunov theorem and the n2 linear equations.

◮ The main ideas of Lyapunov stability are particularly useful innonlinear systems, where the goal is to show that

d

dt‖x(t) − ǫ‖2 < 0

for all x(t) ∈ Rn where ǫ is an equilibrium of the system.

◮ In nonlinear systems, local Lyapunov stability is also useful, i.e.

d

dt‖x(t) − ǫ‖2 < 0 for all x(t) ∈ Xǫ ⊂ R

n.


ECE504: Lecture 8

Lyapunov Stability: DT-LTI Systems

In DT-LTI systems, we only need to analyze the equilibrium at the origin,so the Lyapunov idea says that we want

v(x[k + 1]) − v(x[k]) < 0

wherev(x[k]) = x⊤[k]P x[k] with positive definite P

is a generalized squared distance as in the CT-LTI case. We can write

v(x[k + 1]) − v(x[k]) < 0

⇔ v(Ax[k]) − v(x[k]) < 0

⇔ x⊤[k]A⊤P Ax[k] − x⊤[k]P x[k] < 0

⇔ x⊤[k](

A⊤P A − P)

x[k] < 0

⇔ −x⊤[k]Qx[k] < 0

Is Q := P −A⊤P A is symmetric? When must v(x[k +1])− v(x[k]) < 0?Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 27 / 30

ECE504: Lecture 8

Lyapunov Theorem: DT-LTI Systems

Recall that, for DT-LTI systems:

A is Schur ⇔ the DT-LTI system is asymptotically stable

Theorem

A is Schur if and only if, for any positive definite Q ∈ Rn×n, there exists a

unique positive definite P ∈ Rn×n satisfying P − A⊤P A = Q.

The proof of this theorem can also be found in the Chen textbookSection 5.4.

The same basic procedure as the CT-LTI case applies here (in general, youhave to solve n2 simultaneous linear equations).


ECE504: Lecture 8

Stability of Time-Varying Systems

◮ Example

A(t) =

[

−1 e2t

0 −1

]

What are the eigenvalues of A(t)? Is this system stable orasymptotically stable?

◮ We can use the fundamental matrix method to compute the CT-STM:

Φ(t, 0) =1

2

[

2e−t et − e−t

0 2e−t

]

Now what do you think about the internal stability of this system?◮ In general, eigenvalue analysis is not useful for time-varying systems.

The concept of Lyapunov stability, however, can still be applied (butthe theorems for LTI systems do not directly apply).

◮ Please read Chen pp. 138-140 for a discussion of the internal stabilityof time-varying systems.


ECE504: Lecture 8

Final Remarks

1. State-space description allows us to analyze the internal stabilityof dynamic systems.

2. General concepts of stability and asymptotic stability apply tolinear, non-linear, time-invariant, and time-varying systems.

3. The concept of local stability applies only to nonlinear systemswith distinct equilibria.

4. Several stability theorems for LTI systems.

5. Lyapunov stability analysis is a general tool that can be appliedto all types of systems.


ece504: lecture 8 - worcester polytechnic institutespinlab.wpi.edu/courses/ece504_2008/8.pdf ·...

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