Stability Liapunov criteria

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Chapter 8

STABILITY ANALYSIS

8.1 Introduction:

For linear time invariant (LTI) systems, the concept of stability is simple and can be

formalised as per the following two notions:

a) A system is stable with zero input and arbitrary initial conditions if the resulting

trajectory tends towards the equilibrium state.

b) A system is stable if with bounded input, the system output is bounded.

In nonlinear systems, unfortunately, there is no definite correspondence between the two

notions. The linear autonomous systems have only one equilibrium state and their

behaviour about the equilibrium state completely determines the qualitative behaviour in

the entire state plane. In nonlinear systems, on the other hand, system behaviour for small

deviations about the equilibrium point may be different from that for large deviations.

Therefore, local stability does not imply stability in the overall state plane and the two

concepts should be considered separately. Secondly, in a nonlinear system with multiple

equilibrium states, the system trajectories may move away from one equilibrium point

and tend to other as time progresses. Thus it appears that in case of nonlinear systems,

there is no point in talking about system stability. More meaningful will be to talk about

the stability of an equilibrium point. Stability in the region close to the equilibrium point

or in the neighbourhood of equilibrium point is called stability in the small. For a larger

region around the equilibrium point, the stability may be referred to as stability in the

large. In the extreme case, we can talk about the stability of a trajectory starting from

anywhere in the complete state space, this being called global stability. A simple physical

illustration of different types of stability is shown in Fig. 8.1

Fig. 8.1 Global and local stability


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Equilibrium State: In the system of equation , a state xe wheref(xe,t) = 0 for

all t is called an equilibrium state of the system. If the system is linear time invariant,

namelyf(x,t) = Ax, then there exists only one equilibrium state if A is non -singular and

there exist infinitely many equilibrium states if A is singular. For nonlinear systems,

there may be one or more equilibrium states. Any isolated equilibrium state can be

shifted to the origin of the coordinates, orf(0,t) = 0, by a translation or coordinates.

8.2 Stability Definitions:

The Russian mathematician A.M. Liapunov has clearly defined the different types of

stability. These are discussed below.

Stability: An equilibrium state xe of the system is said to be stable if for each

real number > 0 there is a real number (,t0) > 0 such that implies

for all t t0. The real number depends on and in general, also

depends on t0. If does not depend on t0, the equilibrium state is said to be uniformly

stable.

An equilibrium state xe of the system of equation is said to be stable in the

sense of Liapunov if, corresponding to each S(), there is an S() such that trajectories

starting in S() do not leave S() as t increases indefinitely.

Asymptotic Stability: An equilibrium state xe of the system is said to be

asymptotically stable, if it is stable in the sense of Liapunov and every solution starting

within S() converges, without leaving S(), to xe as t increases indefinitely.

Asymptotic Stability in the Large: If asymptotic stability holds for all states from

which trajectories originate, the equilibrium state is said to be asymptotically stable in the

large. An equilibrium state xe of the system is said to be asymptoticallystable in the large, if it is stable and if every solution converges to xe as t increases

indefinitely. Obviously a necessary condition for asymptotic stability in the large is that

there is only one equilibrium state in the whole state space.

The above said definitions are represented graphically in Fig. 8.2

Fig. 8.2 Liapunovs Stability

Instability: An equilibrium state xe is said to be unstable, if for some real number > 0

and any real number > 0, no matter how small, there is always a state x0 in S() such

that the trajectory starting at this state leaves S().


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8.3 Stability by the Method of Liapunov

Russian mathematician A.M. Liapunov has proposed a few theorems for the study of

stability of the system. The most popular among this is called the Second Method of

Liapunov or Direct Method of Liapunov. This method is very general in its

formulation and can be used to study of stability of linear or nonlinear systems. The

method is called direct method as it does not involve the solution of the system

differential equations and stability information is available without solving the equations

which is definitely an advantage for nonlinear systems. The stability information

obtained by this method is precise and involved no approximation.

First Method of Liapunov: The first method of Liapunov, though rarely talked about, is

essentially a theorem stating the conditions under which system stability information can

be inferred by examining the simplified equations obtained through local linearization.

This theorem is applicable only to autonomous systems.

8.4 Sign Definiteness

Let V(x1, x2, x3, . Xn) be a scalar function of the state variables x1, x2, x3, .., xn.

Then the following definitions are useful for the discussion of Liapunovs second method.

8.4.1 Scalar Functions:

A scalar function V(x) is said to be positive definite in a region if V(x) > 0 for all

nonzero states x in the region and V(0) = 0.

A scalar function V(x) is said to be negative definite in a region if V(x) < 0 for all

nonzero states x in the region and V(0) = 0.

A scalar function V(x) is said to be positive semi-definite in a region if it is positive for

all states in the region and except at the origin and at certain other states, where it is

zero.

A scalar function V(x) is said to be negative semi-definite in a region if it is negative

for all states in the region and except at the origin and at certain otherstates, where it iszero.

A scalar function V(x) is said to be indefinite if in the region it assumes both positive

and negative values, no matter how small the region is.

Examples:

8.4.2 Sylvesters Criteria for Definiteness:

A necessary and sufficient condition in order that the quadratic form xTAx, where A is an

nxn real symmetric matrix, be positive definite is that the determinant of A be positive

and the successive principal minors of the determinant of A be positive.


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i.e.,

A necessary and sufficient condition in order that the quadratic form xT

Ax, where A is annxn real symmetric matrix, be negative definite is that the determinant of A be positive if

n is even and negative if n is odd, and the successive principal minors of even order be

positive and the successive principal minors of odd order be negative.

i.e.,

A necessary and sufficient condition in order that the quadratic form xT

Ax, where A is annxn real symmetric matrix, be positive semi-definite is that the determinant of A be

singular and the successive principal minors of the determinant of A be nonnegative.

i.e.,

A necessary and sufficient condition in order that the quadratic form xTAx, where A is an

nxn real symmetric matrix, be negative semi-definite is that the determinant of A be

singular and all the principal minors of even order be nonnegative and those of odd orders

be non positive.

i.e.,

Example 8.1: Using Sylvesters criteria, determine the sign definiteness of the following

quadratic forms

Successive principal minors are

Hence the given quadratic form is positive definite.


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Example 8.2:


Hence, the given quadratic form is indefinite

Example 8.3:


Hence the quadratic form is negative definite.

8.5 Second Method of Liapunov

The second method of Liapunov is based on a generalization of the idea that if the system

has an asymptotically stable equilibrium state, then the stored energy of the system

displaced within the domain of attraction decays with increasing time until it finally

assumes its minimum value at the equilibrium state. The second method of Liapunov

consists of determination of a fictitious energy function called a Liapunov function. The

idea of the Liapunov function is more general than that of energy and is more widely

applicable. Liapunov functions are functions of x1, x2, x3, .., xn. and t. We denoteLiapunov functions V(x1, x2, x3, .., xn, t) or V(x,t) or V(x) if functions do not include

t explicitly. In the second method of Liapunov the sign behaviours of V(x,t) and its time

derivative give information on stability, asymptotic stability or instability of theequilibrium state under consideration at the origin of the state space.

Theorem 1: Suppose that a system is described by , wheref(0,t) = 0 for all t.

If there exists a scalar function V(x,t) having continuous first partial derivatives and

satisfying the following conditions.

1. V(x,t) is positive definite

2.

Then the equilibrium state at the origin is uniformly asymptotically stable. If in addition,then the equilibrium state at the origin is uniformly

asymptotically stable in the large.

A visual analogy may be obtained by considering the surface . This is a

cup shaped surface as shown. The constant V loci are ellipses on the surface of the cup.

Let be the initial condition. If one plots trajectory on the surface shown, the

representative point x(t) crosses the constant V curves and moves towards the lowest

point of the cup which is the equilibrium point.


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Fig. 8.3 Energy function and movement of states

Theorem 2: Suppose that a system is described by , wheref(0,t) = 0 for all t

0. If there exists a scalar function V(x,t) having continuous first partial derivatives andsatisfying the following conditions.

1. V(x,t) is positive definite

2.

3. does not vanish identically in t t0 for any t0 and any x0 0,

where denotes the trajectory or solution starting fromx0 at t0.

Then the equilibrium state at the origin of the system is uniformly asymptotically stable in

the large.

If however, there exists a positive definite scalar function V(x,t) such that is

identically zero, then the system can remain in a limit cycle. The equilibrium state at the

origin, in this case, is said to be stable in the sense of Liapunov.

Theorem 3: Suppose that a system is described by wheref(0,t) = 0 for all t

t0. If there exists a scalar function W(x,t) having continuous first partial derivatives and

satisfying the following conditions

1. W(x,t) is positive definite in some region about the origin.

2. is positive definite in the same region, then the equilibrium state at theorigin is unstable.


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Example 8.4: Determine the stability of following system using Liapunovs method.

Let us choose

Then

This is negative definite. Hence the system is asymptotically stable.


Let us choose

Then

This is a negative semi definite function. If is to be vanish identically for t t0, then

x2 must be zero for all t t0.

This means that vanishes identically only at the origin. Hence the equilibrium state

at the origin is asymptotically stable in the large.


Let us choose

Then

This is an indefinite function.

Let us choose another

Then

This is negative semi definite function. If is to be vanish identically for t t0, then

x2 must be zero for all t t0.

This means that vanishes identically only at the origin. Hence the equilibrium state

at the origin is asymptotically stable in the large.

Example 8.7: Consider a nonlinear system governed by the state equations

Let us choose

Then


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Therefore, for asymptotic stability we require that the above condition is satisfied. The

region of state space where this condition is not satisfied is possibly the region of

instability. Let us concentrate on the region of state space where this condition is

satisfied. The limiting condition for such a region is . The dividing

lines lie in the first and third quadrants and are rectangular hyperbolas as shown in

Figure. 8.4. In the second and fourth quadrants, the inequality is satisfied for all values of

x1 and x2. Figure 8.4 shows the region of stability and possible instability. Since thechoice of Liapunov function is not unique, it may be possible to choose another Liapunov

function for the system under consideration which yields a larger region of stability.

Conclusions:

1. Failure in finding a V function to show stability or asymptotic stability or instabil ity

of the equilibrium state under consideration can give no information on stability.

2. Although a particular V function may prove that the equilibrium state underconsideration is stable or asymptotically stable in the region , which includes this

equilibrium state, it does not necessarily mean that the motions are unstable outside

the region .

3. For a stable or asymptotically stable equilibrium state, a V function with the required

properties always exists.

8.6 Stability Analysis of Linear Systems

Theorem: The equilibrium state x = 0 of the system given by equation is

asymptotically stable if and only if given any positive definite Hermitian matrix Q (or

positive definite real symmetric matrix), there exists a positive definite Hermitian matrix

P (or positive definite real symmetric matrix) such that . The scalarfunction is a Liapunov function for the system.


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Hence, for the asymptotic stability of the system of , it is sufficient that Q be

positive definite. Instead of first specifying a positive definite matrix P and examining

whether or not Q is positive definite, it is convenient to specify a positive definite matrix

Q first and then examine whether or not P determined from is also

positive definite. Note that P being positive definite is a necessary condition.

Note:

1. If does not vanish identically along any trajectory, then Q may be

chosen to be positive semi definite.

2. In determining whether or not there exists a positive definite Hermitian or real

symmetric matrix P, it is convenient to choose Q = I, where I is the identity matrix.

Then the elements of P are determined from and the matrix P is

tested for positive definiteness.

Example 8.8: Determine the stability of the system described by

Assume a Liapunov function V(x) = XTPX

-2P12 = -1

P11P12P22 = 0

2P122P22 = -1

Solving the above equations,

P11 = 1.5; P12 = 0.5; and P22 = 1

1.5 > 0 and det(P) > 0. Therefore, P is positive define. Hence, the equilibrium state at

origin is asymptotically stable in the large.

The Liapunov function V(x) = XTPX


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Example 8.9: Determine the stability of the equilibrium state of the following system.

4P11 + (1j)P12 + (1+j)P21 = 1

(1j)P11 + 5P21 + (1j)P22 = 0

(1 + j)P11 + 5P12 + (1 +j)P22 = 0

(1j)P12 + (1 + j)P21 + 6P22 = 1


P11 = 3/8; P12 = - (1 + j)/8; P12 = - (1 - j)/8; P22 = 1/4

P is positive definite. Hence the origin of the system is asymptotically stable.

Example 8.10: Determine the stability of the system described by

Assume a Liapunov function V(x) = XTPX

-2P11 + 2P12 = -1

-2P115P12 + P22 = 0

-4P128P22 = -1


P11 = 23/60; P12 = -7/60; and P22 = 11/60


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23 > 0 and det(P) > 0. Therefore, P is positive define. Hence, the equilibrium state at

origin is asymptotically stable in the large.

Example 8.11: Determine the stability range for the gain K of the system given below.

In determining the stability range of k, we assume u = 0.

Let us choose a positive semi definite real symmetric matrix

This choice is permissible since cannot be identically equal to zeroexcept at the origin. To verify this, note that

being identically zero implies that x3 is identically zero. If x3 is identically zero,

then x1 must be must be zero since we have 0 = - kx10. If x1 is identically zero, then x2must also be identically zero since 0 = x2. Thus is identically zero only at the

origin. Hence we may use the Q matrix defined by a psd matrix.

Let us solve,

i.e., -k p13k p13 = 0

-k p23 + p112p12 = 0

-k p33 + p12p13 = 0

P122p22 + p122p22 = 0

P132p23 + p22p23 = 0

P23p33 + p23p33 = -1

Solving the above equations, we get

For P to be positive definite, it is necessary and sufficient that 12 2k > 0 and k > 0 or

0 < k


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8.7 Stability Analysis of Nonlinear Systems

In a linear free dynamic system if the equilibrium state is locally asymptotically stable,

then it is asymptotically stable in the large. In a nonlinear free dynamic system, however,an equilibrium state can be locally asymptotically stable without being asymptotically

stable in the large. Hence implications of asymptotic stability of equilibrium states of

linear systems and those of nonlinear systems are quite different.

If we are to examine asymptotic stability of equilibrium states of non linear systems,

stability analysis of linearised models of nonlinear systems is completely inadequate. We

must investigate nonlinear systems without linearization. Several methods based on the

second method of Liapunov are available for this purpose. They include Krasovskiis

method for testing sufficient conditions for asymptotic stability of nonlinear systems,

Schultz Gibsons variable gradient method for generating Liapunov functions etc.

Krasovskiis Method:

Consider the system defined by , where x is an n-dimensional vector. Assumethat f(0) = 0 and that f(x) is differentiable with respect to xi where, I = 1,2,3,..,n. The

Jacobian matrix F(x) for the system is

Define is the conjugate transpose of F(x). If the

Hermitian matrix is negative definite, then the equilibrium state x = 0 is

asymptotically stable. A Liapunov function for this system is . If in

addition , then the equilibrium state is asymptotically stable

in the large.

Proof:

If is negative definite for all x 0, the determinant of is nonzero everywhereexcept at x = 0. There is no other equilibrium state than x = 0 in the entire state space.

Since f(0) = 0, f(x) 0 for x 0, and , is positive definite. Note that

We can obtain as


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. Therefore the

equilibrium state is asymptotically stable in the large.

Example 8.14: Using Krasovskiis theorem, examine the stability of the equilibrium state

x = 0 of the system given by

This is a negative definite matrix and hence the equilibrium state is asymptotically stable.

. Therefore the equilibrium state

is asymptotically stable in the large.


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CONTROLLERS

The controller is an element which accepts the error in some form and decides the proper

corrective action. The output of the controller is then applied to the system or process.

The accuracy of the entire system depends on how sensitive is the controller to the error

detected and how it is manipulating such an error. The controller has its own logic tohandle the error. The controllers are classified based on the response and mode of

operation. On the basis of mode of operation, they are classified into Continuous and

Discontinuous controllers. The discontinuous mode controllers are further classified as

ON-OFF controllers and multiposition controllers. Continuous mode controllers,

depending on the input-output relationship, are classified into three basic types named as

Proportional controller, Integral controller and Derivative controller. In many practical

cases, these controllers are used in combinations. The examples of such composite

controllers are Proportional Integral (PI) controllers, Proportional Derivative (PD)

controllers and Proportional - IntegralDerivative (PID) controllers.

The block diagram of a basic control system with controller is shown in Figure. The error

detector compares the feedback signal b(t) with the reference input r(t) to generate anerror. e(t) = r(t)b(t).

Proportional Controller:

In the proportional control mode, the output of the controller is proportional to the error

e(t). The relation between the error and the controller output is determined by a constant

called proportional gain constant denoted as KP. i.e. p(t) = KP e(t).

Though there exists linear relation between controller output and the error, for zero error

the controller output should not be zero as this will lead to zero input to the system or

process. Hence, there exists some controller output Po for the zero error. Therefore

mathematically the proportional control mode is expressed as P(t) = KP.e(t) + Po.

The performance of proportional controller depends on the proper design of the gain KP.As the proportional gain KP increases, the system gain will increase and hence the steady

state error will decrease. But due to high gain, peak overshoot and settling time increases

and this may lead to instability of the system. So, compromise is made to keep steady

state error and overshoot within acceptable limits. Hence, when the proportional

controller is used, error reduces but can not make it zero. The proportional controller is

suitable where manual reset of the operating point is possible and the load changes are

small.


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Integral Controller:

We have seen that proportional controller can not adapt with the changing load

conditions. To overcome this situation, integral mode or reset action controller is used.

In this controller, the controller output P(t) is changed at a rate which is proportional to

actuating error signal e(t). Mathematically the integral controller mode is expressed as. The constant Ki is called integral constant. The output from

the controller at any instant is the area under the actuating error curve up to that instant.

If the error is zero, the controller output will not change. The integral controller is

relatively slow controller. It changes its output at a rate which is dependent on the

integrating time constant, until the error signal is cancelled. Compared to the

proportional controller, the integral control requires time to build up an appreciable

output. However it continues to act till the error signal disappears. Hence, with the

integral controller the steady state error can be made to zero. The reciprocal of integral

constant is known as integral time constant Ti. i.e., Ti = 1/Ki.

Derivative Controller:

In this mode, the output of the controller depends on the rate of change of error with

respect to time. Hence it is also known as rate action mode or anticipatory action mode.

The mathematical equation for derivative controller is . Where Kd

is the derivative gain constant. The derivative gain constant indicates by how much

percentage the controller output must change for every percentage per second rate of

change of the error. The advantage of the derivative control action is that it responds to

the rate of change of error and can produce the significant correction before the

magnitude of the actuating error becomes too large. Derivative control thus anticipates

the actuating error, initiates an early corrective action and tends to increase stability of the

system by improving the transient response. When the error is zero or constant, the

derivative controller output is zero. Hence it is never used alone.

PI Controller:

This is a composite control mode obtained by combining the proportional mode and

integral mode. The mathematical expression for PI controller is

The transfer function is given by

The advantage of this controller is that the one to one correspondence of proportional

controller and the elimination of steady state error due to integral controller. Basically

integral controller is a low-pass circuit.

The PI Controller has following effects on the system.


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1. It increases the order of the system

2. It increases the type of the system

3. It improves the steady state accuracy.

4. It increases the rise time so response become slow.

5. It filters out the high frequency noise

6. It makes the response more oscillatory.

PD Controller:

This is a composite control mode obtained by combining the proportional mode and

derivative mode. The mathematical expression for PI controller is


The PD Controller has following effects on the system.

1. It increases the damping ratio and reduces overshoot.

2. It reduces the rise time and makes response fast.

3. It reduces the settling time

4. The type of the system remains unchanged.

5. Steady state error remains unchanged.In general it improves transient part without affecting steady state.

PID Controller:

This is a composite control mode obtained by combining the proportional mode, integral

mode and derivative mode. The mathematical expression for PI controller is



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This is to note that derivative control is effective in the transient part of the response as

error is varying, whereas in the steady state, usually if any error is there, it is constant and

does not vary with time. In this aspect, derivative control is not effective in the steady

state part of the response. In the steady state part, if any error is there, integral control

will be effective to give proper correction to minimize the steady state error. An integral

controller is basically a low pass circuit and hence will not be effective in transient part of

the response where error is fast changing. Hence for the whole range of time response

both derivative and integral control actions should be provided in addition to the inbuiltproportional control action for negative feedback control systems.

Example: The figure shows PD controller used for the system. Determine the value of Tdsuch that the system will be critically damped. Calculate the settling time.

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Stability Liapunov criteria