Idea Transcript
18.12.2015
Design Methods of Linear Control Systems
Visiting Professor Mehmet Dal Department of Electrical and Computer Engineering at TUM, Munchen, Germany 2015
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Contents Introduction to Automatic Control System Control System Design Methods Simulation and Implementation
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System modelling and analysis To analysis and controls design for a linear time-invariant (LTI) system, it can be represented in several ways, devided into two groups: 1) Time domanin (t) • Differential equation • Difference equation for discrete-time domain • State variable form 2) Frequency Domain • Transfer function • Block diagram or flow graph • Impulse response Each description can be converted to others, and provides different approach to analysis and controls design TUM, Germany, 2015 by m.dal
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Order of the systems and their properties • The order of a system reflects its number of energy strorage elements. • A serial RC circuit can be cosidered a simple example of First order system (low pass filter) and a serial RLC circuit is of a Second order. • For both systems, the input voltage and the output voltage are selected as the input and the output, respectively.
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First order system (series RC circuit) From the KVL 1) differential equation of the circuit du (t ) 1 i(t ) u (t ), i(t ) C c dt C di(t ) 1 du(t ) R i(t ) dt C dt Ri(t )
2) state space equation
3) transfer function
duc (t ) 1 1 uc (t ) u (t ) RC dt RC u x
RsI ( s )
x
B
A
dx(t ) Ax(t ) Bu (t ) dt
4) block diagram
1 I ( s ) su ( s ) C
1 ) I (s) u (s) Cs 1 uc ( s ) 1 Cs u ( s) R 1 RCS 1 Cs
(R
block simplification fomula TUM, Germany, 2015 by m.dal
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Responce of first order system 1) Time response
• •
0, u (t ) Vs ,
t t0 t t0
1 t Vs (1 e RC ), t t 0 u c (t ) Vs , t 0, t 0
A first order system corresponds to delay element, like a Low Pass Filter (LPF) The time constant of the circuit RC • Bandwidth (BW) for low pass system is defined as frequency, where the magnitude of voltage gain dropes by a factor of .1 / 2 0.707 ( 3dB )
uc 0.707Vs ) log 20( ) 3dB u Vs If Bode diagram is ploted for a=2, it shows that bandwidth is equal to pole magnatude, BW a 1 / log 20(
•
2) Frequance response Normalized transfer function of the system gives its characteristic equation and poles
a sa Ch( s ) s a 0
Gain
G (s)
s a 1 / RC f(Hz) TUM, Germany, 2015 by m.dal
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a and u ( s ) 1 / s sa then the output uc ( s ) is
G( s)
Unite step response uc(t)
The system transfer fynction and unite step input defined as follows
1 a a s s a s( s a) its time domen response can be find from the table of Laplas transform uc ( s ) u ( s )G ( s )
t(s)
uc (t ) 1(1 e at ), for u (t ) 1
Tr 2.2 and Td 0.69 A good notes from the text book
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SECOND ORDER SYSTEMS Many useful systems are of second order, for high order systems oftenly the dominant (low frequency) pole pairs are analiyzed to approximate the system with a second order transfer function. Therefore, the transfer function of second order system is very important
the general form of second order system transfer function n2 n2 C ( s) G ( s) 2 2 2 R( s ) s 2s n s 2 n s n2 The denominator is the Characteristic polynomial H ( s) s 2 2 n n2
the poles of the system from equating H(s) to 0 are s1, 2 j j ( ) 2 n
2
s1, 2 n j n 1 2
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Step respose of second order system The rise time tr, which is the time required for the step response to rise from 0.1 to 0.9 of its steady-state value. The settling time ts is the time required for the signal to effectively reach its steady-state value. For the pure exponential response t r 2.2 , t s 4 or 5
unit step response: C ( s)
n2 1 , R( s ) 1 / s, unite step input 2 s s 2 n s n2
M p 1 100e Tp
1 2
n ( 1 2
POS 100e
1 2
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Effects of Damping ratio s1, 2 n j n 1 2
and n n 1 2 *Underdamped case if 0 < ξ ωn we have two real poles generated One the these poles will move towards the left on the real axis and the other to the right. The system response is now very slow, and it is said to be overdamped. There are no oscillations. Another important property of a series RLC circuit is the impedance transfer function. Z (s)
u ( s ) LCs 2 RCs 1 i( s) Cs
If we let s = jω (i.e. the resonant frequency), and substitute this into
Clearly the magnitude of this expression has a minimum value when the imaginary term is zero. Therefore: 2 LC 1 0
1
LC
n
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Quality Factor Quality Factor: Another important measure of resonant second order circuits
• the total energy is: • the energy dissipated over a period To is 1 2 LI m L Q 2 2 2 f 0 1 R 2 LI m R 2 f0 Q
1 L using n R C
1 LC
If Q = 0.5 then which is the same expression for the resistance when the circuit is critically damped
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Test of second order system analysis G( s)
u c (s) 1 u (s) LCs 2 RCs 1
Build a transfer function block in Matlab/simulink, find paramters and simulate the model for step input to explore the three different cases: 1) undamped, 2) underdaped and 3) overdamped showing the system output, uc(s), on the scope screen
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Feedback control system design objectives Properties of control systems: • Stability For the bounded input signal, the output must be bounded and if input is zero then output must be zero then such a control system is said to be stable system • Performance
-Dynamic stability -Accuracy
Additional considerations: • Robustness (insensitivity to parametervariation) to models (uncertainties and nonlinearities) to disturbances, and to noises • Cost of control • System reliability
Dynamic overshooting Oscillation duration Steady-state error -Speed of (Transient) response
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Requirement of Good Control System Bandwidth: An operating frequency range decides the bandwidth of control system. Bandwidth should be large as possible for frequency response of good control system. Speed: It is the time taken by control system to achieve its stable output. A good control system possesses high speed. The transient period for such system is very small. Oscillation: A small numbers of oscillation or constant oscillation of output tend to system to be stable.
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Brief view of control techniques • Classical control: Proportional-integral-derivative (PID) control, developed in 1940s and used for control of industrial processes. Examples: chemical plants, commercial aeroplanes. • Optimal control: Linear quadratic Gaussian control (LQG), Kalman filter, H2 control, developed in 1960s to optimize a certain ‘cost index’ and boomed by NASA Apollo Project. • Adaptive control: Uses online identification of the process parameters, thereby obtaining strong robustness properties. Adaptive control was applied for the first time in the aerospace industry in the 1950s. • Robust control: H∞ control, developed in 1980s & 90s to achieve robust performance and/or stability in the presence of small modeling errors. Example: military systems. • Nonlinear control: Currently hot research topics, developed to handle nonlinear systems with high performances. Examples: military systems such as aircraft, missiles. • Intelligent control: Predictive control, neural networks, fuzzy logic, machine learning, evolutionary computation and genetic algorithms, researched heavily in 1990s, developed to handle systems with unknown models. Examples: economic systems, social systems, human systems. TUM, Germany, 2015 by m.dal
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Methods of Analysis and Design in Linear Control System Cosidered • Mathematical Models of Systems – Laplace transforms and transfer functions – State-space model • Feedback Characteristics and Performance – Time-domain performance specifications – Stability, transient and steady-state responses – Ziegler–Nichols algorithm • Model based analytical design – Full state-feedbck (pole placement) • Complex-domain method – Root locus method for analysis and design of control systems • Frequency-domain method – Frequency-domain performance specifications – Bode plot diagrams for analysis and design of control systems • Design of control system
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Three terms control, Prortional Integral Derivative (PID) A PID control algorithm involves three separate parameters namely; Kp, Ki, Kd constant gain values. • The proportional value determines the reaction to the current error, • The integral value determines the reaction based on the sum of recent errors, and • The derivative value determines the reaction based on the rate at which the error has been changing, By tuning the three constants in the PID controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms • Steady state error, e • Overshoots, Mp • The degree of system oscillation, which corresponds to settling time ts [Ogata-2002, Dorf and Bishop-2005].
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Characteristics of P, I, and D Controllers Determining the values for Ki, Kp and Kd with the correlations in the table may be used. But changing one of these variables can change the effect of the other two, therefore these values are not exactly accurate, because Ki, Kp and Kd are dependent of each other. For this reason, the table should only be used as a reference when you determine parameters Ki, Kp and Kd with the use of the trial-error method is used.
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Model of PMDC motor with transfer function block di Ri u k e dt e
(1)
d Bω k t i t L J dt t
( 2)
L
b
J
d kt i t L dt t
e
if B is omitted
e
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State space model of PM DC motor k b J J d dt i k k i x L L x
A
1 L u 0
;
1 0 i y Cx
B
Simulation model for state variables i and ω di Ri 1 1 u k e dt L L L e
(1)
d b 1 1 ω kt i t L dt J J t J
( 2)
b
e
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Cascade controlled system (DC motor drives) • An electric motor drive system can have three cascade controller • Design should be started from the festest control loop, in this case, it is the most inner loop, (current control loop). The current is more faster then machanical variables, speed and position.
Mp %5 for speed, t s 4Tc 20ms for current control loop T s m
J 7 s and ei 0, e 0 B TUM, Germany, 2015 by m.dal
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PI controller model based design 1) Designing current loop control
If the back-emf is assumed as disturbance of the system, then it can be neglected, eb=0, the open loop system transfer function K pi (Ti s 1) K pi 1 L I ( s) Gc .G s Fio ( s ) for Ti Te 1.1ms Fio ( s ) R E ( s) Ti s R (Te s 1) Ls Closed loop transfer functions Fic ( s )
Fio ( s ) I * ( s) 1 L L Fic ( s ) s , Tc L I ( s) 1 Fio ( s) K pi K pi s 1 K pi
t s 4Tc K pi 4
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L ts
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Speed loop control 2) Designing speed loop control
* For an optimal second-order control system in set-point control is given by Fmc ( s )
Fmo ( s )
K p (T s 1) T s
K p k t Js(Ti s 1)
K p 1 1 1/ B 1 k , Tm T B / J kt J Ti s 1 t B Ti s 1 (Tm s 1) s B
* Open loop gain Kpω is set for critical damping 1 / 2
K p k t K p k t Js(Ti s 1) 1 K p k t / J J Fmcl ( s ) K p k t Js(Ti s 1) K p k t K k J / 2 p t 1 1 Ti s s K p k t Js(Ti s 1) J TUM, Germany, 2015, m.dal
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Speed loop control K p k t
K p k t J
Fcl ( s )
Ti s 2 s
2 n
n2
K p k t
1 K pi Ti L
K p K pi k t JL
J
n2 JTi , K pi 2 1 K p k t K pi Ti s 2 n s n2 s2 s Ti JL
K pi J K p 2 4 Lk t
ts
4
n
8L K pi
M p 1 100e Tp
1 2
n ( 1 2
POS 100e
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1 2
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Ziegler–Nichols Methods • Most useful when a mathematical model of the plant is not available. • Proportional‐integral‐derivative (PID) control framework is a method to control uncertain systems • Many different PID tuning rules available • Transfer function of a PID controller
• The three term control signal
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The S-shaped step response Ziegler–Nichols Tuning Formula (first method) • The S-shaped curve may be characterized by two parameters: delay time L and time constant T • The transfer function of such a plant may be approximated by a firstorder system with a transport delay
Table 1.
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Ziegler‐Nichols PID Tuning (Second Method) (Use the proportional controller to force sustained oscillations) In this method, the closed-loop system behavior is observed. A P-controller is used to tune the system towards oscillation boundary. The gain is increased until the system is on the oscillation boundary. Then the output of the system oscillates with constant amplitude and frequency. The parameters of the controller are calculated according to table 2. Table 2.
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State-Feedback Control (pole-placement control ) The method of feed-backing all the state variables to the input of the system through a suitable feedback matrix in the control strategy is known as the full-state variable feedback control technique. In this approach, the poles or eigenvalues of the closed loop system can be placed arbitrarily at the specified location. Placing the poles or eigenvalues of the closed-loop system at specified locations arbitrarily if and only if the system is controllable. Pole placement is easier if the system is given in controllable form. Thus, the aim is to design a feedback controller that will move some or all of the open-loop poles of the measured system to the desired closedloop pole location as specified. Hence, this approach is also known as the pole-placement control (or Pole Assignment, Pole Allocation) design. TUM, Germany, 2015 by m.dal
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State‐space representation dx Ax Bu x(0) x 0 dt
dx Ax Bu , dt y Cx
x1 x x 2 Rn xn
u1 u u 2 Rm um
x R n , u R, y R
State feedback control law
u Kx,
k11 k1n K R mxn k m1 k mn
K is the controller gain matrix Requires measurement of all state variables measurable
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Control Design using Pole Placement •
Objectives – Choose eigenvalues of closed-loop system – Decrease response time of open-loop stable system – Stabilize open-loop unstable system The state feedback controller is designed using pole placement technique via Ackermann’s formula
the characteristic polynomial for this closed-loop system is the determinant of [sI - (A-BK)] x*(t) = (A − BK)x(t) y=Cx
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Introducing the Reference Input
• Since the matrices A and BK are both 2x2 matrices, there should be 2 poles for the system. • By designing a full-state feedback controller, we can move these two poles anywhere we want them. • first try to place them at -5+j and -5-j (note that this corresponds to a ξ = 0.98 which gives 0.1% overshoot and a sigma = 5 which leads to a 1 sec settling time).
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The basic difference with root locus design Drawback: One of the major disadvantage of state feedback controller design by using only the pole-placement is the introduction of a large steady-state error. In order to compensate this problem, an integral control is added where it will eliminate the steady-state error in responding to a step input. In brief the pole assignment technique is somewhat similar to the root locus method in that a closed loop poles are placed at desired locations. The basic difference is that in root locus design only the dominant closed loop poles are placed at the desired locations, while in the pole assignment technique all the closed loop poles are placed at the desired locations. [Ogata-1998, Ogata-2002, Dorf and Bishop-2005].
State feedback Control design • Closed‐loop system dx Ax Bu Ax BKx ( A BK )x A BK R nxn dt
• Design objective
– Choose K such that sA‐BK) are placed at the desired locations – Closed‐loop characteristic equation
s ( A BK ) s n an 1s n 1 a1s a0 0
– Desired closed‐loop characteristic equation
s n n 1s n 1 1s 0 0
– Equate powers to determine K
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Controllability (Reachability) • Eigenvalues can be placed arbitrarily if and only if system is controllable • Single input (m = 1) – Controllability matrix
Co B AB A n 1B R nxn
– System is controllable if Co is nonsingular
• Multiple inputs (m > 1) – Controllability matrix
Co R nxm
– System is controllable if rank(Co) = n
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Illustrative Example • Linear model dx 1 dt 2
1 1 x u Ax B u 4 0
• Open‐loop stability – si(A) = ‐0.438, ‐4.56 – Origin is a stable steady state
• Controllability 1 1 1 1 AB 2 4 0 2 1 1 Co B AB 0 2
rank (Co) n 2 0 Co is non-sigular so that the system is completly controllable
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Illustrative Example contin. • Characteristic equation 1 1 1 A BK k1 2 4 0
1 k1 1 k 2 k2 4 2
s ( A BK ) sI ( A BK )
s 1 k1 2
1 k2 s4
( s 1 k1 )( s 4) ( 1 k 2 )( 2) s 2 (5 k1 ) s 4 k1 2 k 2 2
• Desired characteristic equation ( s 0.3)( s 0.4) s 2 0.7 s 0.12
• Controller gains
5 k1 0.7 4 k1 2 k 2 2 0.12
k1 4.30 k 2 7.66
State‐space model of DC motor d B 1 1 ω kt i t L dt J J t J
(1)
di Ri 1 1 u k e dt L L L e
( 2)
e
b
ke kt k
k b 1 J J d L u dt i k k 0 x i L L B A
x
1 0 ; y C x i
For a control system defined in state-space form and depending on the pole assignment the control signal will be
u Kx
Where the control signal u is determined by the instantaneous states x1=i and x2=ω The rest of the design presuderes follows the preious illustrative example TUM, Germany, 2015 by m.dal
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Hardware Test Setup
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PM DC motor and its Parameters Vdc (V) 12 N (rpm) 5800 Te (Nm) 94e-3 Tmax (Nm) 28.4e-3 Imax (A) 1.5 Ra (Ω ) 2.5 La (H) 300e-6 Ld (H) 3e-3 J kg m2 17.6e-7 B (Nms/rad) 1.41e-6 Tc (Nm) 0 Ke Vs/rad 19.5e-3 Km 19.5e-3 Rd (Ω ) 0.69 Jb1 kg m2 8.11e-6
Department of EnergyTechnology, AAU 2012 by m.dal
Terminal nominal voltage No_load speed Stall torque Max continious torque Max continious current Terminal resistance Rotor inductance inductivity of added coil Rotor inertia Viscous damping conctant Coloumb friction back-EMF constant torque constant resistance of added coil inertia of swing-wheel
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Content of DC motor Interface Block
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Building A DC motor model Lab Procedures for Model Building • Build a dynamic model for a separately PM DC motor. • A suggested system block diagram is shown in given block diagram • Refer to your lecture notes for details. • Print the model including block diagrams of all subsystems. The DC motor has the nameplate data and parameters given the table in slide 49. The load torque TL is assumed to be zero.
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Group study
Simulation methods ========= tasks
Grup1 Separate Simulink blocks
Grup2 Func, mux, integrator, sum blocks
Grup3 Sys func. Block (m‐file)
Grup4 Embedded Matlab func. blocks
building and simulation of IM designing controller and PWM Integration and simu. of drive scheme.
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Control design tutorial By Prof. Bill Messner, Carnegie Mellon University and Prof. Dawn Tilbury, University of Michigan.
http://www.engin.umich.edu/class/ctms/matlab42/index.htm
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Disturbance Observer Compensation for the disturbance torque on the rotor shaft makes a drive robust against load changes and unmodeled torques.
te t L J
d Bω dt
t L kt i J
tˆL
d Bω dt
tl
Rotor te, m
2 g ktni J n sω gJ n g J n g ktni sg sg sg
Note: The ideal derivative is not realizable in digital implementation so that the differentiation task is performed by a Low Pass Filter (LPF). M. Nakao, K. Ohnishi, and K. Miyachi, “A robust decentralized joint control based on interference estimation,” In Proc. IEEE Int. Conf Robotics and Automat., vol. 1, pp. 326-331, 1987. TUM, Germany, 2015 by m.dal
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