Basics of Dynamical and Control System (CS )cse.iitkgp.ac.in/~soumya/cps/slides/8-basic-control-2.pdf · Basics of Dynamical and Control System (CS|) Soumyajit Dey CSE, IIT Kharagpur

Basics ofDynamical andControl System

(CS—)

Soumyajit DeyCSE, IIT

Kharagpur

What is Stability??

Stability Analysisfrom Closed LoopTransfer functionin S-Plane

Stability Analysisof Linear andNonlinear Systems

What is Stability ?? Stability Analysis from Closed Loop Transfer function in S-Plane Stability Analysis of Linear and Nonlinear Systems

Basics of Dynamical and Control System(CS—)

Soumyajit DeyCSE, IIT Kharagpur

Soumyajit Dey CSE, IIT Kharagpur Basics of Dynamical and Control System (CS—)


(CS—)


Kharagpur

What is Stability??




Table of Contents

1 What is Stability ??

2 Stability Analysis from Closed Loop Transfer function inS-Plane

3 Stability Analysis of Linear and Nonlinear Systems



(CS—)


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What is Stability??




The Concept of Stability: Physical Significance

The notion of stability is very old and has a clearintuitive meaning.

Let us take an ordinary pendulum and put it in thelowest position, in which it is stable. Now, put it in theutmost upper position where it is unstable.

Stable and unstable situations can be seen everywhere -in mechanical motion, in technical devices, in medicaltreatment (stable or unstable state of the patient) andso on.



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What is Stability??




The Concept of Stability: Physical SignificanceThe concept of stability can be illustrated by a cone placedon a plane horizontal surface:

Figure: Conceptual description of linear system stability.

Ref: nptel.ac.in



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What is Stability??




The Concept of Stability: Why Important?

(a) Opening day of theTacoma Narrows Bridge,Tacoma, Washington, July 1,1940. The bridge was foundto oscillate whenever thewind used to blow.

(b) The 1940 TacomaNarrows Bridge collapsing ina 42 miles per hour (68km/h) gust on November 7,1940

Figure: An example of unstable system in real life.

Ref: https://en.wikipedia.org/wiki/



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What is Stability??




Definitions of Stability

A system is Stable when its natural response goes tozero as time approaches infinity.

A system is Unstable when its natural response goes toinfinity as time approaches infinity.

A system is Marginally Stable when its natural responseremains constant or oscillates within a bound.

Ref: nptel.ac.in



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What is Stability??




Definitions of Stability

BIBO stability: A system is said to be BIBO stable if for anybounded input, its output is also bounded.

Thus for any bounded input the output either remain constant ordecrease with time.

Absolute stability: Stable /Unstable

Relative stability: Degree of stability (i.e. how far from instability).Relative system stability can be measured by observing the relativereal part of each root. In the following diagram r2 is relativelymore stable than the pair of roots labeled as r1.

Figure: Example of relative system stability.

Ref: www.calvin.edu



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What is Stability??




Table of Contents






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What is Stability??




Stability: S-Plane and Transient Response

A necessary and sufficient condition for a feedback system tobe stable is that all the poles of the system transfer functionhave negative real parts.

Ref: nptel.ac.in



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What is Stability??




Stability Analysis from Closed Loop Transferfunction

Stable systems have closed-loop transfer functionswhose poles reside only in the left half-plane.

Unstable systems have closed-loop transfer functionswith at least one pole in the right half plane and/orpoles of multiplicity greater than one on the imaginaryaxis.

Marginally Stable systems have closed-loop transferfunctions with only imaginary axis poles of multiplicity 1and poles in the left half-plane.



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What is Stability??




Stability Analysis

Let us consider the transfer function of a closed-loop system:

G (S) =C (S)

R(S)=

∑mi=0 ci s

m−i∑ni=0 ri sn−i

Conditions for Stability

Necessary condition for stability:– All coefficients of R(s) have the same sign.

Necessary and sufficient condition for stability:– All poles of G(s) reside in the left-half-plane (LHP)

i .e. R(s) 6= 0 for Re[s] ≥ 0



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What is Stability??




Stability Analysis

Necessary condition for stability

R(s) = r0sn + r1s

n−1 + ...+ rn−1s + rn

= r0(s + p1)(s + p2)...(s + pn)

= r0sn + r0(p1 + p2 + ...+ pn)sn−1

+r0(p1p2 + ...+ pn−1pn)sn−2

...+ r0(p1p2...pn)

−p1 to −pn are the poles of the system.

Therefore, given a system to be stable:

All poles of the system must have negative real parts.

The coefficients of the polynomial should have the same sign.

Examples

R(s) = s3 + s2 + s + 1 can be stable or unstable

R(s) = s3 − s2 + s + 1 is unstable



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What is Stability??




Table of Contents






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What is Stability??




Stability of Linear Time Invariant (LTI) Systems

Let us consider a general linear (time-invariant) system givenby:

x = Ax and x(0) = X0, x ∈ <m

It may represent the closed or open loop system whereA ∈ <m×m.

Eigenvalues of A are the “Poles” of the given system.

We can define the nature of the solution, withoutsolving the system model.

The nature of the solution is governed only by thelocations of its poles.



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Stability Properties of a Linear SystemConsidering the linear system defined earlier, and for each eigenvalue λof A, suppose that mλ denotes the algebraic multiplicity of λ and dλthe geometric multiplicity of λ.

We can conclude the following:

The system is asymptotically stable if and only if A is a stabilitymatrix; i.e., every eigenvalue of A has a negative real part.

The system is neutrally stable if and only if

Every eigenvalue of A has a nonpositive real part, andAt least one eigenvalue has a zero real part, anddλ = mλ for every eigenvalue λ with a zero real part.

The system is unstable if and only if

Some eigenvalue of A has a positive real part, orThere is an eigenvalue λ with a zero real part anddλ < mλ.

Ref: https://www.ru.ac.za @Claudiu C. Remsing, 2006.



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What is Stability??




Motivation for Stability Analysis of Non-LinearSystems

Eigenvalue analysis concept is not suitable for nonlinearsystems.

Non-linear systems can have multiple equilibrium pointsand limit cycles.

Stability behaviour of nonlinear systems need not bealways global (unlike linear systems).



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What is Stability??




Stability of Non-Linear Systems

Basic Concepts

Let us consider a nonlinear dynamical system∑

defined by

x = F (t, x), x ∈ <m

where x(·) is a curve in the state space <m and F is avector-valued mapping having components Fi , i = 1, 2, ...,m.

Here, we will assume that the components Fi arecontinuous and satisfy standard conditions.

From a geometric point of view, the right-hand side(RHS) F can be interpreted as a time-dependentvector field on <m.

If the functions Fi do not depend explicitly on t, thensystem

∑is called autonomous (or time-independent);

otherwise, nonautonomous (or time-dependent).



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What is Stability??





Equilibrium states

If F (t, c) = 0 for all t, then c ∈ <m is said to be anequilibrium (or critical) state.

It follows that (for an equilibrium state c) if x(t0) = c , thenx(t) = c for all t ≥ t0. Thus solution curves starting at cremain there.

The intuitive idea of stability in a dynamical setting is thatfor “small” perturbations from the equilibrium state at sometime t0, subsequent motions t → x(t), t ≥ t0 should not betoo “large”.



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What is Stability??





Equilibrium statesConsider a ball resting in equilibrium on a sheet of metal bent intovarious shapes with cross-sections as shown below:

If frictional forces can be neglected, then small perturbations lead to :

oscillatory motion about equilibrium (case (i)) ;

the ball moving away without returning to equilibrium (case (ii));

oscillatory motion about equilibrium, unless the initialperturbation is so large that the ball is forced to oscillate about anew equilibrium position (case (iii)).




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What is Stability??




Stability in the Sense of LyapunovLet us Consider the general nonautonomous system:

x = f (x , t), x(t0) = x0 ∈ <n

where the control input u(t) = h(x(t), t), has been combined into thesystem function f . Without loss of generality, let us assume that theorigin x = 0 is the system equilibrium of interest.

This system is said to be stable in the sense of Lyapunov with respectto the equilibrium x∗ = 0, if for any ε > 0 and any initial time t0 ≥ 0,there exists a constant, δ = δ(ε, t0) > 0, such that

||x(t0)|| < δ ⇒ ||x(t)|| < ε for all t ≥ t0

This stability is illustrated by the following figure:

Figure: Geometric meaning of stability in the sense of Lyapunov.

Ref: www.ee.cityu.edu.hk/ gchen/pdf/C-Encyclopedia04



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What is Stability??




Asymptotic StabilityConsider the same general nonautonomous system:

x = f (x , t), x(t0) = x0 ∈ <n

This system is said to be asymptotically stable about its equilibriumx∗ = 0, if it is stable in the sense of Lyapunov and, furthermore, thereexists a constant, δ = δ(t0) > 0, such that

||x(t0)|| < δ ⇒ ||x(t)|| → 0 as t →∞

This stability can be visualized by the following figure:

Figure: Geometric meaning of the asymptotic stability.




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What is Stability??




Asymptotic Stability: Classification

Uniform Asymptotic Stability

The asymptotic stability is said to be uniform if the existingconstant δ is independent of t0 over [0,∞).

Global Asymptotic Stability

The asymptotic stability is said to be global if theconvergence, ||x || → 0, is independent of the initial statex(t0) over the entire spatial domain on which the system isdefined (e.g., when δ =∞).




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What is Stability??




Exponential StabilityConsidering the same general nonautonomous system defined earlier,the equilibrium state is said to be exponentially stable if

it is stable in the sense of Lyapunov and,

there exists two positive constants c and σ, such that

||x(t0)|| < δ ⇒ ||x(t)|| ≤ ce−σt

This stability is visualized by the following figure:

Figure: Geometric meaning of the exponential stability.




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What is Stability??




Asymptotic and Exponential Stabilities

We can conclude that exponential stability implies asymptotic stability,and asymptotic stability implies the stability in the sense of Lyapunov,but the reverse need not be true.

Examples

1. Let us take a system which has output trajectory x1(t) = x0sin(t);it is stable in the sense of Lyapunov about 0, but is notasymptotically stable.

2. A system with output trajectory x2(t) = x0(1 + t − t0)−1 isasymptotically stable (so also is stable in the sense of Lyapunov) ift0 < 1 but is not exponentially stable about 0.

3. A system x3(t) = x0e−t is exponentially stable (hence, is both

asymptotically stable and stable in the sense of Lyapunov).




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What is Stability??




Lyapunov TheoryThe so-called “direct” method of Lyapunov in relation to thenonlinear autonomous dynamical system

∑given by

x = F (x), x(0) = x0 ∈ <m; F (0) = 0.

To deal with the (nonautonomous) case some changes are required:

x = F (t, x), x(t0) = x0

Lyapunov theory is used to determine the stability nature of theequilibrium state (at the origin) of system

∑without obtaining

the solution x(·).

The main idea is to generalize the concept of energy V for aconservative system in mechanics, where a well-known result statesthat an equilibrium point is stable if the energy is minimum.

Hence, V is a positive function which has V negative in theneighbourhood of a stable equilibrium point.




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What is Stability??




Lyapunov function

Definition

We define a Lyapunov function V : <m → < as follows:

V and all its partial derivatives ∂V∂xi

are continuous;

V is positive definite (PD); i.e.,

i. V (x) ≥ 0 for all xii. V (x) = 0 if and only if x = 0iii. all sublevel sets of V are bounded, which is equivalent

to V (x)→∞ as x →∞



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What is Stability??




Lyapunov function

The (directional) derivative of V with respect to the vector field, F canbe written as

V =∂V

∂xF =

[∂V∂x1· · · ∂V

∂xm

]F1

...Fm

=∂V

∂x1F1 +

∂V

∂x2F2 +

∂V

∂xmFm.

A Lyapunov function V for the defined system∑

is termed as

strong if the derivative V is negative definite; i.e., V (0) = 0 andV (x) < 0 for x 6= 0 such that ||x || ≤ k.

weak if the derivative V is negative semi-definite; i.e., V (0) = 0and V (x) ≤ 0 for all x such that ||x || ≤ k.




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What is Stability??




The Lyapunov stability theorems

The two basic theorems of Lyapunov are:

Lyapunovs First Theorem

Suppose that there is a strong Lyapunov function V forsystem

∑. Then system

∑is asymptotically stable.

Lyapunovs Second Theorem

Suppose that there is a weak Lyapunov function V forsystem

∑. Then system

∑is stable.




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What is Stability??




Instability theorem

Let us consider an autonomous dynamical system andassume an equilibrium point to be at X = 0. Now if theLyapunov function(V : D → <) for the system has thefollowing properties:

i. V (0) = 0

ii. ∃X0 ∈ <n (arbitrarily close to X = 0), such thatV (X0) > 0

iii. V > 0 ∀X ∈ U, where the set U is defined by

U = {X ∈ D : ||X || ≤ ε and V (X ) > 0}

Under all these conditions, the equilibrium state X = 0 issaid to be unstable.



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What is Stability??




Example

Let us Consider a unit mass suspended from a fixed support by a spring,z being the displacement from the equilibrium. If first the spring isassumed to obey Hookes law, then the equation of motion is

z + kz = 0 where k is the spring constant.

Now if we assume x1 → z and x2 → z , the equation of motion can bedefined as {

x1 = x2

x2 = −kx1Since the system is conservative, the total energy

E =1

2kx2

1 +1

2x22

is a Lyapunov function(V ) and it is easy to observe that

E = kx1x2 − kx2x1 = 0

Hence, by Lyapunov’s Second Theorem the origin of the system isstable.



Basics of Dynamical and Control System (CS )cse.iitkgp.ac.in/~soumya/cps/slides/8-basic-control-2.pdf · Basics of Dynamical and Control System (CS|) Soumyajit Dey CSE, IIT Kharagpur

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