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Control system design essentialsThe root-locus method to
controller design
Spacemaster Preparation CourseControl system design basics
Lei Ma
Department of Computer Science VII: Robotics and
TelematicsUniversity of Würzburg
10.10.2007
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
References
Ogata,K. Modern control engineering. Prentice Hall.Brogan,W.L.
Modern control theory. Prentice Hall.Lunze,J. Regelungstechnik 1,2.
Springer.Goodwin,G.C et al. Control system design. Prentice
Hall.Jeffrey, A. Mathematics for Engineers and Scientists.Chapman
& Hall/CRC 2004Stroud, K.A., Booth, D.J. Engineering
Mathematics.Industrial Press 2001Dettman, J.W. Introduction to
Linear Algebra andDifferential Equations. Dover Publications
1986.Golub, G.H., Van Loan, C.F. Matrix Computations (JohnsHopkins
Studies in Mathematical Sciences). The JohnsHopkins University
Press 1996
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
The principal goal of control
The fundamental goal of control engineering is to
findtechnically, environmentally, and economically feasible ways
ofacting on systems to control their outputs to desired values,thus
ensuring a desired level of performance. (Goodwin)
Stability
Disturbance rejection
Regulation and tracking
Dynamics
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Typical unit-step response of a second-order system
h(t)
t
Mp
tr
tp
ts
td
Allowable tolerance (5% or
2% of the desired output)
Delay time tdRise time tr (underdamped 0~100%, overdamped
10%~90%)
Peak time tpMaximum overshoot Mp
Settling time ts
Figure: Typical unit-step response of a second-order system
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Some popular structures of control
r(t)K(s) G(s)
H(s)
e(t) u(t) y(t)
_controller plant
sensor
K(s) G(s)e(t) u(t) y(t)
_controller plant
Standard control loop Unit-feedback control
r(t)
K1(s) G(s)
H(s)
e(t) u(t) y(t)
_Controller 1 plant
sensor
r(t)
K2(s)
_
Controller 2
Inner loop
Outer loop
K(s) G(s)
H(s)
e(t) u(t) y(t)
_controller plant
sensor
K(s)r(t)
Standard control loop
with pre-filter
Pre-filter
Cascad control
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
The transfer functions
K(s) G(s)e(t) u(t) y(t)
_controller plant
Unit-feedback control with disturbances
and noisy measurement
r(t)
d(t)
disturbances
n(t)
noise
Open-loop TF.: G0(s) = K (s)G(s)
Closed-loop TF.: Gr (s) =G0(s)
1+G0(s), (D(s) = N(s) = 0)
Closed-loop TF. according to disturbances:Gd(s) = 11+G0(s) ,
(R(s) = N(s) = 0)
Closed-loop TF. according to noise:Gn(s) = −
G0(s)1+G0(s)
= −Gr (s), (R(s) = D(s) = 0)
Y (s) = Gr (s)R(s) + Gd(s)D(s) + Gn(s)N(s)
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
The steady-state error
Steady-state error of step response with d(t) = 0 and n(t) =
0:limt→∞ e(t) = lims→0 s 1s
E(s)R(s) = lims→0
11+G0(s)
.
G0(s) does play an important role! In case n ≥ q, let
G0(s) = kslbqsq+bq−1sq−1+...+b1s+b0
an−l sn−l+an−l−1sn−l−1+...+a1s+a0, the follows are true:
For l = 0, G0(0) = k0 = kb0a0
and limt→∞ e(t) = 11+k0 ;
For l > 0, lims→0 G0(s) = ∞ and limt→∞ e(t) = 0.
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
Dynamic performance of a second-order system
h(t)
t
Mp
tr
tp
ts
td
Allowable tolerance (5% or
2% of the desired output)
Delay time tdRise time tr (underdamped 0~100%, overdamped
10%~90%)
Peak time tpMaximum overshoot Mp
Settling time ts
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
ObjectivesThe standard control loopBehaviors of the control
loopThe controllers
The PID controller
kp
kI/s
kDs
kp 1/sTI
TDs
=E(s)E(s) U(s) U(s)
KPID(s) = kP +kIs
+ kDs = kP(1 +1
sTI+ sTD)
Rise time Overshoot Settling time S-S errorkP Decrease Increase
Small change DecreasekI Decrease Increase Increase EliminatekD
Small change Decrease Decrease Small change
Table: Impact of P, I, D on dynamic performances
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Open-loop and closed-loop system
G(s)
H(s)
R(s) Y(s)
Open-loop system:G(s)H(s)
Closed-loop transfer function:
Y (s)R(s)
=G(s)
1 + G(s)H(s)
The characteristic equation:
1 + G(s)H(s) = 1 + K(s + z1)(s + z2) . . . (s + zm)(s + p1)(s +
p2) . . . (s + pn)
= 0
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Idea: Use the poles and zeroes of the open loop system
todetermine the properties of the closed loop system when
ONEparameter is changing.
In control theory, the root locus is the locus of the poles of
atransfer function as the system gain K is varied on someinterval.
The root locus is a useful tool for analyzing single inputsingle
output (SISO) linear dynamic systems. A system isstable if all of
its poles are in the left-hand side of the s-plane.(Wikipedia)
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Angle and magnitude conditions
the characteristic equation indicates that G(s)H(s) =
−1.Therefore the values of s that fulfill both the angle
andmagnitude conditions are the roots of the
characteristicequation, or the closed-loop poles.
Angle condition:∠G(s)H(s) = ±180◦(2k + 1), k = 0, 1, 2, . .
..
Magnitude condition: |G(s)H(s)| = 1.
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Characters of root loci
Symmetry: As all roots are either real or complexconjugate pairs
so that the root locus is symmetrical to thereal axis.
Number of branches: The number of branches of the rootlocus is
equal to the number of poles of the open-looptransfer function.
Locus start and end points: The locus starting points are atthe
open-loop poles and the locus ending points are at theopen-loop
zeros. n − m branches end at infinity. Thenumber of starting
branches from a pole and endingbranches at a zero is equal to the
multiplicity of the polesand zeros, respectively. A point at
infinity is considered asan equivalent zero of multiplicity equal
to n − m.
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
General rules of constructing root loci (Ogata)
Locate the poles and zeros of G(s)H(s) on the s plane.Determine
the root loci on the real axis.Determine the asymptotes of root
loci.Find the break-away and break-in points.Determine the angle of
departure (angle of arrival) of theroot locus from a complex pole
(at a complex zero).Find the points where the root loci may cross
the imaginaryaxis.Taking a series of test points in the broad
neighborhood ofthe origin of the s plane.Determine closed-loop
poles.
For details see Ogata!
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Example
G(s)
H(s)
R(s) Y(s)
G(s) = Ks(s+1)(s+2) , H(s) = 1
Angle condition: ∠G(s) = ∠ Ks(s+1)(s+2) =
−∠s − ∠(s + 1) − ∠(s + 2) = ±180◦
(2k + 1), k = 0, 1, 2, . . .
Magnitude condition: |G(s)| =∣
∣
∣
Ks(s+1)(s+2)
∣
∣
∣= 1.
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Example
Locate the poles and zeros of G(s)H(s) on the s plane: 3poles,
no zeros. Root loci have 3 branches.
Determine the root loci on the real axis using
anglecondition.
Determine the asymptotes of root loci
Angles of asymptotes = ±180◦
(2k+1)3 , k = 0, 1, 2, . . .;
Finding the point where the asymptotes intersect the realaxis:s
= −
∑ni=1 pi−
∑qi=1 zi
n−q .
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Example
Find the break-away point where the two branches startfrom s = 0
and s = −1 leave the real axis and go to thecomplex plane: dkds =
0.
Determine the points where the root loci cross theimaginary axis
by setting s = jω.
Choose a test point in the broad neighborhood of the jωaxis and
the origin.
Draw the root loci.
Determine a pair of dominant complex-conjugateclosed-loop poles
such that ζ = 0.5.
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Typical distributions of open-loop poles and zeros andthe root
loci
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Typical distributions of open-loop poles and zeros andthe root
loci
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Outline
1 Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
2 The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus
Lei Ma Pre-Course: Control system design basics
-
Control system design essentialsThe root-locus method to
controller design
The ideaConstructing root lociController design based on root
locus
Typical design procedure using root locus
Derive locations of the dominant pare of poles based onthe
design criteria;
Choose a controller kK̂ (s);
Draw the root loci for the open-loop systemkK̂ (s)G(s)H(s);
Determine value of the gain k which gives the dominantpare of
poles. Determine locations of other poles;
Simulate the system. Choose complexer K̂ (s) if the
designcriteria are not satisfied.
controller: kK̂ (s).
Lei Ma Pre-Course: Control system design basics
Control system design essentialsObjectivesThe standard control
loopBehaviors of the control loopThe controllers
The root-locus method to controller designThe ideaConstructing
root lociController design based on root locus