lec 7. higher order systems, stability, and routh stability criteria higher order systems stability...
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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)
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Nonstandard 2nd Order Systems
So far we have been focused on standard 2nd order systems
Non-unit DC gain:
Extra zero:
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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
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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.
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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
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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)
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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)
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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
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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
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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.
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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
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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
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A Necessary Condition for Stability
If is stable (assume a00)
Then have the same sign, and are nonzero
Example:
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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”
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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
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Example
Determine the stability of
Check by Matlab command: roots([1 2 3 4 5])
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Stability vs. Parameter Range+
Determine the range of parameter K so that the closed loop system is stable
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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
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Another Example
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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: