Lec 7. Higher Order Systems, Stability,and Routh Stability Criteria

• Higher order systems

• Stability

• Routh Stability Criterion

• Reading: 5-4; 5-5, 5.6 (skip the state-space part)

Nonstandard 2nd Order Systems

So far we have been focused on standard 2nd order systems

Non-unit DC gain:

Extra zero:

Effect of Extra Zero

standard form

Under any input, say, unit step signal, the response of H(s) is

unit-step response of standard 2nd order system

Unit-Step Response (=0.4,=1,n=1)

Step Response

Time (sec)

Am

plit

ud

e

0 5 10 15-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

The introduction of the extra zero affects overshoot in the step response.

Higher Order Systemsn-th order system:

It has n poles p1,…,pn and m zeros z1,…,zm

Factored form:

As in second order systems, locations of poles have important implications on system responses

Distinct Real Poles Case

Suppose the n poles p1,…,pn are real and distinct

Partial fraction decomposition of H(s):

where 1,…,n are residues of the poles

Unit-impulse response:

Unit-step response:

The transient terms will eventually vanish if and only if all the poles p1,…,pn are negative (on the LHP)

(parallel connection of n first order systems)

Distinct Poles (may be complex)

Suppose that the n poles p1,…,pn are distinct (may be complex)

Partial fraction decomposition of H(s):

Unit-impulse response:

Unit-step response:

(parallel connection of q first order systems and r second order systems)

The transient terms will eventually vanish if and only if all the poles p1,…,pn have negative real part (on the LHP)

Remarks• Effect of poles on transient response

– Each real pole p contributes an exponential term

– Each complex pair of poles contributes a modulated oscillation

– The magnitude of contribution depends on residues, hence on zeros

• Stability of system responses– The transient term will converge to zero only if all poles are on the LHP

– The further to the left on the LHP for the poles, the faster the convergence

• Dominant poles– Poles with dominant effect on transient response

– Can be real, or complex conjugate pair

Example of Dominant Poles

Step Response

Time (sec)

Am

plit

ud

e

0 1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

Stability of Systems

• One of the most important problems in control (ex. aircraft altitude control, driving cars, inverted pendulum, etc.)

• System is stable if, under bounded input, its output will converge to a finite value, i.e., transient terms will eventually vanish. Otherwise, it is unstable

• A system modeled by a transfer function

is stable if all poles are strictly on the left half plane.

Problems Related to StabilityStability Criterion: for a given system, determine if it is stable

Stabilization: for a given system that may be unstable, design a feedback controller so that the overall system is stable.

+

plantcontroller

How to Determine Stability

Transfer function is stable

All roots of are on the LHP

Method 1: Direct factorization

Method 2: Routh’s Stability Criterion

Determine the # of roots on the LHP, on the RHP, and on j axis without having to solve the equation.

“stable polynomial”

Advantage: • Less computation• Works when some of the coefficients depend on parameters

A Necessary Condition for Stability

If is stable (assume a00)

Then have the same sign, and are nonzero

Example:

Routh’s Stability Criterion

Step 1: determine if all the coefficients of

have the same sign and are nonzero. If not, system is unstable

Step 2: arrange all the coefficients in the follow format

“Routh array”

Routh’s Stability Criterion (cont.)

Step 3: # of RHP roots is equal to # of sign changes in the first column

Hence the polynomial is stable if the first column does not change sign

Routh array

Example

Determine the stability of

Check by Matlab command: roots([1 2 3 4 5])

Stability vs. Parameter Range+

Determine the range of parameter K so that the closed loop system is stable

Special Case IThe first term in one row of the Routh array may become zero

Example:

Replace the leading zero by

Continue to fill out the array

Let and let N+ be the # of sign changes in the first column

Let and let N- be the # of sign changes in the first column

Another Example

Special Case IIAn entire row of the Routh array may become zero

Example:

Auxiliary polynomial

No sign changes in the first column, hence no additional RHP roots

Roots of auxiliary polynomial are roots of the original polynomial

See textbook pp. 279 for a more complicated example.

Derivative of auxiliary polynomial: