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Control system design essentials The root-locus method to controller design

Spacemaster Preparation Course Control system design basics

Lei Ma

Department of Computer Science VII: Robotics and Telematics University of Würzburg

10.10.2007

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Outline

1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

Lei Ma Pre-Course: Control system design basics

Control system design essentials The 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 2004 Stroud, K.A., Booth, D.J. Engineering Mathematics. Industrial Press 2001 Dettman, J.W. Introduction to Linear Algebra and Differential Equations. Dover Publications 1986. Golub, G.H., Van Loan, C.F. Matrix Computations (Johns Hopkins Studies in Mathematical Sciences). The Johns Hopkins University Press 1996

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The controllers

Outline

1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The controllers

The principal goal of control

The fundamental goal of control engineering is to find technically, environmentally, and economically feasible ways of acting 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 essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The 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 td Rise time tr (underdamped 0~100%, overdamped 10%~90%)

Peak time tp Maximum 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 essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The controllers

Outline

1 Control system design essentials Objectives The standard control loop Behaviors of the control loop The controllers

2 The root-locus method to controller design The idea Constructing root loci Controller design based on root locus

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The 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 essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The 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 essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The controllers

Outline

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The 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

1 1+G0(s)

.

G0(s) does play an important role! In case n ≥ q, let

G0(s) = ksl bqsq+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 = k b0 a0

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 essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The controllers

Outline

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The 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 td Rise time tr (underdamped 0~100%, overdamped 10%~90%)

Peak time tp Maximum overshoot Mp

Settling time ts

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

Objectives The standard control loop Behaviors of the control loop The controllers

The PID controller

kp

kI/s

kDs

kp 1/sTI

TDs

= E(s)E(s) U(s) U(s)

KPID(s) = kP + kI s

+ kDs = kP(1 + 1

sTI + sTD)

Rise time Overshoot Settling time S-S error kP Decrease Increase Small change Decrease kI Decrease Increase Increase Eliminate kD 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 essentials The root-locus method to controller design

The idea Constructing root loci Controller design based on root locus

Outline

Lei Ma Pre-Course: Control system design basics

Control system design essentials The root-locus method to controller design

The idea Constructing root loci Controller 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 essentials The root-locus method to controller design

The idea Constructing root loci Controller design based on ro