Automation Systems
Systematic approach to the control system design
Jakub Mo˙zaryn, PhD, Eng.
Institute of Automatic Control and Robotics
Warszawa, 2016
Control System Design
Step 1
1. Establish the control goals.
Natural and technological processes
Natural processes
The physical and chemical changes of the state of matter, that are taking place without human intervention.
weather changes,
water movement in rivers, tectonic movements, chemical processes in the human body.
Technological processes
The processes designed by man using devices that are constructed by him, to obtain the intended changes in the state of matter.
temperature changes in the greenhouse,
water level changes in the tanks in chemical plants, movements of robots.
During the course there will be discussed issues related to technological processes.
System description
Example 1: Many modern devices employ a roating disc eg. CD player, hard drive. The goal is to design a system for rotating disc control that will ensure that the actual rotation speed will be within a specified percentage of required speed.
Example 2: Autopilot in car is being developed to operate autonomously for a long period of time. Proposed autopilot is tasked to drive an empty road at constant speed and follow it as acurately as possible. It should also consider lim- itations as signs and traffic regulations.
Step 2
2. Identify the variables to be controlled.
Example - autopilot in cars
Example - autopilot in cars
Step 3
3. Write the specifications.
Specifications
Specifications
Specifications is set of all relevant information concerning details, work and requirements to be satisfied by a designed control system.
Examples:
Type of the process → technology, modelling.
Type, range and required precision of the input and output signals
→ choice of sensors and actuators.
Working conditions of the system → limitations, noise, external disturbances.
Operation of the system → rejection of exteral disturbances, following the reference trajectory.
Required system behaviour (response) → precision, settling time, overshoot, robustness.
Price.
Step 4
4. Establish the system configuration.
Control System
Control System
Control System is the interconnection of components forming a system configuration that shall provide a desired system response (behaviour).
Single Input Single Output - SISO system
Multiple Input Multiple Output - MIMO system
Open-loop control system
Examples:
Toaster
Cofee vending machine
Signals yr(t) - desired output response
u(t) - control signal x (t) - input signal y (t) - output signal
Closed-loop control system
Examples:
Temperature control in greenhouse
Water level control in a tank Automatic pilot
Signals u(t) - control signal x (t) - input signal y (t) - output signal yr(t) - desired output response
ym(t) - measured output e(t) = yr(t) − ym(t) - error signal
Closed-loop control system
Closed-loop feedback control system
Control system that tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using their difference as a means of control. Therefore:
error signal e(t) is amplified,
controller causes the actuator to modulate the process in order to reduce the error e(t).
Feedback concept
A closed-loop control system uses a measurement of the output and feedback of this signal to compare it with the desired output (reference or command).
Advantages of the closed-loop control system over open-loop control system:
rejects external disturbances d (t),
improves measurement noise n(t) attenuation.
Autonomous rover
Figure :A rover using an embedded computer in the feedback loop. (Photo by R.H. Bishop.)
Step 5
5. Obtain the model of the process, the actuator and the sensor.
Mathematical modelling for control systems
Real processes, and thus control systems, have nonlinear properties:
turbulences,
multiple stable states, hysteresis,
energy losses due to friction.
In practice, to simplify the mathematical description, there is carried linearization, enabling the formulation of the approximate description of a linear phenomenon, in vincinity of the operating point (this point cor- responds to the most nominal or averaged operating conditions of the system).
description of the phenomenon in the form of differential equations, linearization,
operational calculus - algebraic equations.
Description of linear models
The basic forms of mathematical description of the dynamical linear system are:
equations of system dynamics in form of differential equations, transfer function,
state space equations.
In the case of the system with one input signal x (t) and one output signal y (t) dynamics equation expresses the relationship between the output signal y (t) and the input signal x (t).
Description of linear models / systems
Linear system
Linear system is the system, which preserves the principle of superposi- tion and principle of homogeneity. It is said that the space of solutions of the equation that satisfies the principle of superposition is a linear space.
Superposition:
y (x1+ x2) = y (x1) + y (x2), oraz y (0) = 0 (1) . Homogenity (implies scale invariance):
y (βx ) = βy (x ), and y (0) = 0 (2) . where: β - constatnt coefficient.
Nonlinear system
The system, which does not preserve the principle of superposition and/or principle of homogeneity
Description of linear models / systems
General form of linear system differential equation:
an
dny dtn+ an−1
dn−1y
dtn−1+ · · · + a0y = bm
dmx dtm + bm−1
dm−1x
dtm−1+ · · · + b0x (3) where: y - output signal, u - input signal, ai, bi - constant coefficients.
Laplace transform
Replacing differential equation with algebraic equation requires transition from the time domain t to the complex plane S .
f (t) ⇔ f (s), where s = c + j ω (4) where: c - real part coefficient, ω - conjugate part coefficient.
Laplace transform
f (s) = L[f (t)] =
∞
Z
0
f (t)e−stdt (5)
Laplace transform is used for an analysis of control systems - multipli- cation by s has the effect of differentiation and division by s has the effect of integration. Analysis of complex roots of a linear equa- tion, may disclose information about the frequency characteristics and the stability of the system.
Laplace transform of the linear systems
Linear system is described by following differential equation andny
dtn+ an−1dn−1y
dtn−1+ · · · + a0y = bmdmx
dtm + bm−1dm−1x
dtm−1+ · · · + b0x (6) using Laplace transform
L dny dtn
= sny (s) − sn−1y (0+) − · · · − yn−1(0+) (7) and assuming that initial conditions are zero, one obtains
L dny dtn
= sny (s) (8)
Laplace transform of the linear system with zero initial conditions take the following form
y (s)(ansn+an−1sn−1+· · ·+a0) = x (s)(bmsm+bm−1sm−1+· · ·+b0) (9)
Transfer function
Transfer function
For continuous-time input signal x (t) and output y (t), the transfer func- tion G (s) is the linear mapping of the Laplace transform of the input, X (s) = L[x (t)], to the Laplace transform of the output Y (s) = L[y (t)] at zero initial conditions:
y (s)(ansn+an−1sn−1+· · ·+a0) = x (s)(bmsm+bm−1sm−1+· · ·+b0) (10) G (s) = y (s)
x (s) =bmsm+ bm−1sm−1+ · · · + b0 ansn+ an−1sn−1+ · · · + a0
(11) Numerator
M(s) = bmsm+ bm−1sm−1+ · · · + b0 (12) Denominator - characteristic equation
N(s) = ansn+ an−1sn−1+ · · · + a0 (13)
Basic dynamical elements - differential equations
y (t) = ku(t) Proportional element
(non-inertial)
Tdy (t)
dt + y (t) = ku(t) First-Order Lag Tdy (t)
dt = u(t), or dy (t)
dt = ku(t) Integrator
y (t) = Tdu(t)
dt Differentiator (ideal)
Tdy (t)
dt + y (t) = Td
du(t)
dt Differentiator (real)
T2d2y (t)
dt + 2ξTdy (t)
dt + y (t) = ku(t)
Second Order Lag, if 0 < ξ < 1
y (t) = u(t − T0) Delay
Basic dynamical elements - transfer functions
G (s) = k Proportional element
(non-inertial) G (s) = k
Ts + 1 First-Order Lag
G (s) = 1
Ts, or G (s) = k
Ts Integrator
G (s) = Tds Differentiator (ideal)
G (s) = Tds
Ts + 1 Differentiator (real)
G (s) = k
T2s2+ 2ξTs + 1
Second Order Lag, if 0 < ξ < 1
G (s) = e−T0s Delay
Controlled process - step response
Step response of the static pro- cesses: 1- first order lag element, 2, 3 – higher order lag elements, 4 – oscillatory, 5 - proportional.
Step response of the astatic pro- cesses: 1- integral element, 2 - integral element with first order lag, 3 - integral element with first order lag and delay.
Stability
Stability
A stable system is a dynamic system witch a bounded response to a bounded input
Stable system, if pushed out a state of equilibrium (considered operating point P) returns to the state of equlibrum (to some state K ) after the termination of factors (disturbaces d ) that pushed the system from a state of equilibrum.
In the case of linear systems, the behavior of the system after the termina- tion of an action, that pushed the system out of the equilibrum position, is characteristic feature of the system and does not depend on the type of the action before it’s termination. (simple analysis)
In the case of nonlinear systems, their behavior as effect of an action, that pushed the system out of the equilibrum point, can depend on the type and magnitude of the action before it’s termination. (complicated analysis)
Stability
There are three types of behavior after pushing a system out of the equilibrum point:
1 A system returns to equilibrium state in the operating point prior to action that pushed system out of balance - asymptotical stability.
2 A system returns to equilibrium state in the operationg point other than one present when action pushed system out of balance - non-asymptotical stability, neutral, marginal.
3 A system doesn’t reach a state of equilibrym - ustability,
instability; a special case of such behavior are sustained oscillations with constant amplitude - system on the border of stability.
Figure : a) unstability, b) stability, c) neutral
Experimental determination of the time characteristics of
controlled process
Models of the static process
The characteristic features of the step response of the higher order lag elements are fixed time gains T1and T2defined by the tangent to the step response at the point of inflection (as given in a picture).
Models of the static process
model 1 - first order lag with delay G (s) = ∆ym(s)
∆u(s) = kob
(Tzs + 1)e−T0s (14) model 2 - Strejc model
G (s) = ∆ym(s)
∆u(s) = kob
(Tzs + 1)n (15) model 3 - Strejc model with delay G (s) = ∆ym(s)
∆u(s) = kob
(Tzs + 1)ne−T0s (16)
First order lag model with delay
Model 1 - Tangent method
T0= T1; Tz = T2 (17)
Model 1 - Secant method
Assumption: The step response of the model in 2 points corresponds with the step response of the process.
P = 0, 5PV → t1; P = 0, 632PV → t2 (18) Using the time equation of step response of the first order lag element:
y (t) = ustk(1 − e−Tt), (19) the following equations are obtained:
T0= t1− t2ln 2
1 − ln 2 , (20)
Tz = t2− T0= t2− t1
1 − ln 2. (21)
Higher order lag elements
Model 2 - Strejc model, G (s) = y (s)
u(s) = 1
(Ts + 1)n (22) n T1/T T2/T T1/T2
1 0 1 0
2 0,282 2,718 0,104 3 0,805 3,695 0,218 4 1,425 4,463 0,319 5 2,100 5,119 0,410 6 2,811 5,699 0,493 Table : Parameters of the higher order lag elements
G (s) = y (s)
u(s) = 1
(Ts + 1)6 (23)
Static processes models - example
Static processes models - example
Astatic process models - identification
Integral element with first order lag Integral element with first order lag and delay
Gob(s) = 1
Tzs(T0s + 1) (24) Gob(s) = 1
Tzse−T0s (25)
Gob(s) = 1
Tzs(T1s + 1)e−T0s (26) Gob(s) = 1
Tzse−(T0+T1)s (27)
Step 6
6. Describe the controller and select the key parameters to be adjusted.
Technical realization of controllers
Figure : Technical realization of controllers
Classification of the controllers - pt. 1
Ctiteria Controller type
Type of the processed signals: analogous digital The way of influence on the object: continous
non-continous Compliance with the law of
superposition: linear
nonlinear
Destination: specialized
universal
Classification of the controllers - pt. 2
Ctiteria Controller type
Type of implementation:
mechanical pneumatic hydraulic electrical
Algorithm of control action:
PID controllers other (LQR, state-space, predictive)
The energy required for operation: direct action indirect action
PID controllers
Control Algorithm
The dynamic properties of controllers are realized by control algorithm.
The most popular control algorithm (95 %) is called PID algorithm (Pro- portional - Integral - Derivative). Its possible to realize simpler algorithms:
P, PI, PD, by setting gains of PID controller (kP, Ti, Td).
PID controller - real
Gr(s) = kp
1 + 1
Tis + Tds Td
kds + 1
Figure : Block diagram of PID controller - paralell realization
Technical realization of PID controllers
Figure :A diagram illustrating the functional characteristics of the industrial PID controller
Selection of controllers
The basic premise when choosing the type of controller is dynamic characteristics of the controlled process.
Figure :Control system
Basic equations, describing the properties of the controlled processes Gob(s) = ∆ym(s)
∆u(s) = kob
Tzs + 1e−T0s, Gob(s) = ∆ym(s)
∆u(s) = 1 Tzse−T0s
Selection of controllers
for ratio T0
Tz
< 0, 1 ÷ 0, 2 → switch controllers (two- three- gain),
for ratio 0, 1 ≤ T0
Tz
< 0, 7 ÷ 1 ÷ 0, 2 → continuous controllers,
for ratio T0
Tz
> 1 → impulse controllers (impulse output signals)
In the case of industrial processes common ratio T0
Tz is in the range of 0, 2 ÷ 0, 7. Therefore, in industrial control systems the most common controllers are continuous, with typical control algorithms P, PI, PD and PID.
Selection of controllers
Analysis of controler action with the process leads to the following conclusions concerning the selection of the control algorithm:
PI algorithm provides good control only for the low frequencies of setpoint changes or disturbancs. Integral action is necessary to obtain zero error in steady state.
PI algorithm provides wider bandwidth than PID algorithm, but poorer performance for the low frequencies of setpoint changes or disturbancs.
Derivative action is recommended for objects with higher order lag (such as thermal processes), because it allows the strong interaction of control even at small deviations. PD controller does not ensure the achievement of zero deviation in steady-state . PID algorithm merges to a certain extent the advantages of PI and PD algrithms.
Step 7
7. Optimize the parameters and analyze the performance.
Selection of controllers
In practice, industrial controllers with continuous algorithm and perfor- mance are commonly used. Their parameters (settings) can be changed (adjusted) within a wide range, so they can work properly with processes with different dynamical properties.
Depending on the requirements of the stability and quality, the controller settings are selected by the special selection procedures.
There are following settings of PID controller:
proportional gain kp= 0, 1 ÷ 100 integral gain Ti = 0, 1 ÷ 3600s derrivative gain Td = 0 ÷ 3600s
Tuning of the controllers
Methods of PID controllers tuning:
Experimental methods - do not allow to achieve certain quality of the control system, eg. Ziegler – Nichols, Pessen, Hassen and Offereissen, Cohen-Coon, ¨Astr¨om – Hagglund .
Tabular methods - determining the set of controller parameters based on the parameters of a mathematical model of the controlled process and the required quality criterion of the control system (like the lowest overshoot, short settling time, problem: often
minimization of different quality index base on the contrary requirements).
Autotuning, eg. relay method.
Tuning of the controllers
Ziegler-Nichols method Type 1:
Controller settings are selected on the basis of the parameters of the closed loop control system, brought to the border of stability (by experimental excitation of the system).
It can be used to controller tuning in the control systems where pro- cesses are described by static and astatic, higher order lag elements.
Type 2:
It can be used to controller tuning in the control systems where processes are described by static higher order lag elements.
Controller settings are selected based on the transient response of the controlled process.
Steady state errors
In a control system with a static object and P or PD control algorithm there are non-zero steady state errors in relation to the disturbances or setpoint changes respectively.
Increasing the proportional gain of P or PD controller reduces the value of static deviations. Reducing the static deviation by increasing the gain kp is usually limited due to the stability of the system. (The system with PD controller reaches the border of stability at higher gain than in the case of the regulator P - as in Ziegler-Nichols method).
Integral action in the controller (PI, PID) provides zero steady state errors in relation to the disturbances or setpoint changes respectively.
Dynamical quality of control system
Requirements related to the transient response in the control systems are determined by a number of indices, generally called dynamic performance quality indicies of the control system. Groups of such indices are:
transient response indices,
indices descibing the frequency plots of the control system - magnitude and phase margins,
integral indices.
Transient response indices
To evaluate the transient response following indices are used:
Maximum error (dynamical): em- the maximum value of error after the step change of disturbance or setpoint.
Settling time: tr - it is the time between the moment of change of the set point w (t), or introduction of disturbances z(t) and the moment when the error e(t) reaches a fixed value inside a boundary
∆e(t) (eg.∆e(t) = |0.05emax|).
Overshoot:
κ =
e2
e1
· 100% (28)
where e1 and e2are the first 2 consecutive biggest errors with opposite signs, assuming steady state value of output y (t) after transient response as the zero level (baseline).
Oscillatory transient response - disturbances
Figure : Oscillatory transient response of the control system to disturbances: a) with non-zero static deviation, b) with zero static deviation
Aperiodic transient response - disturbances
Figure :Aperiodic transient response of the control system to disturbances: a) with non-zero steady state error, b) with zero steady state error
Oscillatory transient response - setpoint
Figure :Oscillatory transient response of the control system to setpoint change: a) with non-zero steady state error, b) with zero steady state error
Aperiodic transient response - setpoint
Figure :Aperiodic transient response of the control system to setpoint change:
a) with non-zero steady state error, b) with zero steady state error
Automation Systems
Systematic approach to the control system design
Jakub Mo˙zaryn, PhD, Eng.
Institute of Automatic Control and Robotics
Warszawa, 2016
