# Chapter 3: Analysis of closed-loop systems

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1 Chapter 3: Analysis of closed-loop systems Control Automático 3º Curso. Ing. Industrial Escuela Técnica Superior de Ingenieros Universidad de Sevilla

2 Control of SISO systems Control around an operation point (u,y) r(t) - e(t) Controller u(t) + u(t) Process y(t) u (Reference is not a deviation variable ) Controller e(t) Controller u(t) Automatic control PI How to chose the value of the controller parameters (Kp and Ti) in a way such that the closed-loop system has an appropriate performance?

3 Control of SISO systems Heuristic design Tuning based on experiments Real system Model Design based on tables Tuning based on a set of experiments and a table that determines the value of each parameter Zieger-Nichols (Chapter 6) Mathematical design The tuning is based on a mathematical analisys of the closed-loop system and provides guaranteed properties Transient response Steady state response Robustness Analitic design techniques Root locus design techniques Loop-shaping design techniques

4 Control of SISO systems r(t) - e(t) Controller u(t) + u(t) Process y(t) u Closed-loop system It is hard to analize the dynamics of the closed-loop system if the process or the controller are nonlinear systems The design techniques studied in this course are based on the linearized model of the system around a given operation point (u,y)

5 Incremental variables model The linearized model can be compared with the incremental variables model for a given operating point which is defined as follows: + System - Incremental variables model (u,y) Assumption: The initial state is the operation point Remark: The model depends on the operating point

6 Incremental variables model Linearized model Assumption: Zero initial conditions. - e(t) Controller u(t) Linearized model around (u,y) Analisys of the closed-loop system. In this case, both the controller and the system are LTI systems. Teoría de sistemas Assumption: The properties of this (simplified) system are similar to the ones of the real closed-loop system if the trajectories are close to the operating point - Speed of the transient - Tracking of time-varying references - Disturbance rejection

7 Teoría de sistemas TEMA. Introducción y fundamentos. Sistemas dinámicos. Conceptos básicos. Ecuaciones y evolución temporal. Linealidad en los sistemas dinámicos. TEMA. Representación de sistemas. Clasificación de los sistemas. Clasificación de comportamientos. Señales de prueba. Descripción externa e interna. Ecuaciones diferenciales y en diferencias. Simulación. TEMA 3. Sistemas dinámicos lineales en tiempo continuo. Transformación de Laplace. Descripción externa de los sistemas dinámicos. Función de transferencia. Respuesta impulsional. Descripción interna de los sistemas dinámicos. TEMA 4. Modelado y simulación. Modelado de sistemas. Modelado de sistemas mecánicos. Modelado de sistemas hidráulicos. Modelado de sistemas eléctricos. Modelado de sistemas térmicos. Linealización de modelos no lineales. Modelos lineales. Álgebra de bloques. Simulación. TEMA 5. Respuesta temporal de sistemas lineales. Sistemas dinámicos lineales de primer orden. Ejemplos. Sistemas dinámicos lineales de segundo orden. Respuesta ante escalón. Sistemas de orden n. TEMA 6. Respuesta frecuencial de sistemas lineales. Función de transferencia en el dominio de la frecuencia. Transformación de Fourier. Representación gráfica de la función de transferencia. Diagramas más comunes. Diagrama de Bode. TEMA 7. Estabilidad. Estabilidad de sistemas lineales. Criterios relativos a la descripción externa de los sistemas dinámicos. Criterio de Routh-Hurwitz. Criterio de Nyquist. Criterios relativos a la descripción interna. Analysis of linear time invariant (LTI) systems

8 Closed-loop transfer function Linear time invariant systems (LTI) Remark: From now on, the variables y and u denote incremental values; that is, the deviation of the input and the output from the operating point Laplace transform (assuming zero initial conditions) Properties used: Linearity, transform of the time derivative Transfer function

9 Proportional term The value of the input u(t) is proportional to the error e(t) Controller u(t) Transfer function Time domain E(s) C(s) U(s) Frequency domain Desing parameter: Kp

10 Integral term The value of the input u(t) is proportional to the error and its integral e(t) Controller u(t) Transfer function Temporal domain E(s) C(s) U(s) Frequency domain Desing parameter: Kp, Ti Lag compensation net Controller with properties similar to the PI

11 Derivative term The value of the input u(t) is proportional to the error and its time derivative e(t) Controller u(t) Transfer function Time domain E(s) C(s) U(s) Design parameters: Kp, Td Frequency domain Lead compensator net Controller with properties similar to the PD

12 PID controller The value of the input u(t) is proportional to the error, its time derivative and its integral e(t) Controller u(t) Widely used in industry

13 e(t) PID controller The value of the input u(t) is proportional to the error, its time derivative and its integral Controller u(t) Transfer function Time domain E(s) C(s) U(s) Design parameter: Kp, Td, Ti Frequency domain Lead-lag compensator Controller with properties similar to the PID

14 Index Closed Loop Transfer Function Tuning a Controller Stability analysis of system Steady-state response of a closed-loop systems Transient response of a stable system. Closed-loop poles and zeros vs. Controller parameters

15 Closed-loop transfer function Block algebra (Ogata 3.3, Tema 3, Teoría de sistemas) S(s) + - S(s) S3(s) = S(s)-S(s) S(s) S(s) = S(s) S3(s) = S(s) Signal sum Signal bifurcation S(s) G(s) S(s)=G(s)S(s) LTI system

16 Closed-loop transfer function Closed-loop system R(s) - E(s) C(s) U(s) G(s) Y(s) R(s) Gbc(s) Y(s) Gbc(s) models how the ouput of the closed-loop system reacts to changes in the references The properties of a controller are defined based on the response of the closed-loop system

17 Other transfer functions Sensor dynamics R(s) + C(s) U(s) G(s) Y(s) - Ym(s) H(s) D(s) Disturbances R(s) + - C(s) U(s) G(s) Gd(s) + + Y(s) Ym(s) H(s)

18 Index Closed Loop Transfer Function Tuning a Controller Stability analysis of system Steady-state response of a closed-loop systems Transient response of a stable system. Closed-loop poles and zeros vs. Controller parameters

19 Controller tuning Obtain a set of controller parameter (C(s)) in order to guarantee that the closedloop systems satisfies a given set of conditions (specifications) Gbc(s) is not defined if parameters of C(s) are not fixed Specifications Stability Raise time (step reference change) Steady state error Specifications on the linearized model are relevant to the behaviour of the real closed-loop system Mathematical design Tuning based on the anlisys of the closed-loop dynamics (which depend on the controller parameters)

20 Example Closed-loop system Closed-loop response? Depends on the value of Kp Three poles that depend on Kp Static gain depens on Kp Reference signal: Step change - Simulations performed in Simulink/Matlab

21 Example Kp=. Kp= Kp= Kp=5

22 Example y(t).5 Kp =., Td =, /Ti = y(t).5.5 Kp =, Td =, /Ti = u(t) u(t) e(t) e(t) e(τ)dτ e(τ)dτ t Respuesta del sistema en BC t Respuesta del sistema en BC Kp =, Td =, /Ti = Kp = 5, Td =, /Ti = y(t) y(t) u(t) u(t) e(t) e(t) e(τ)dτ e(τ)dτ t Respuesta del sistema en BC t Respuesta del sistema en BC

23 Example Kp =., Td =, /Ti = y(t) u(t) e(t).5 e(τ)dτ t Respuesta del sistema en BC

24 Example.5 Kp =, Td =, /Ti = y(t) u(t) e(t).5 e(τ)dτ t Respuesta del sistema en BC

25 Example Kp =, Td =, /Ti = y(t) u(t) e(t) e(τ)dτ t Respuesta del sistema en BC

26 Example Kp = 5, Td =, /Ti = y(t) u(t) e(t) e(τ)dτ t Respuesta del sistema en BC

27 Types of behaviour Unit step response Clasification of the output signal y(t) depending on the input signal Unit Step (the most widely used). Ramp. Sinusoidal. Provides information about the dynamic proterties of the system Model expressed in error variables (u,y) Assumption: Initial conditions in the operating point. Unit Step: Behaviours: Overdamped Underdamped Unstable Oscilatory

28 Overdamped Types of behaviour.9.8 Step Response Delay L Gain K Raise Time ts.7 Amplitude K Raise Time: Time to reach 63% of the steady state value. Delay: Time of reaction for the output with respect to a change in the input... Gain: Quotient of the output and the input values. 5 5 L ts Time (sec)

29 Types of behaviour Underdamped Mp K Delay L Gain K Raise Time ts Peak Time tp Settling Time te Overshoot Mp Raise Time: Time to reach the steady state value for the first time. Peak Time: Time to reach the maximum value. Settling Time: Time to confine the output within a band of 5% arounf the steady state value. Overshoot: Percentage Increment of the peak value with respect to the steady state value.. ts tp te

30 Unstable Types of behaviour

31 Index Closed Loop Transfer Function Tuning a Controller Stability analysis of system Steady-state response of a closed-loop systems Transient response of a stable system. Closed-loop poles and zeros vs. Controller parameters

32 Stability (Chapter 7. Stability) Stability Criterion: Gbc(s) is stable if and only if all poles are located on the left-half complex plane. Closed loop poles are the roots of (depend on C(s)) A closed-loop system might become unstable if the controller is not properly designed. Example: Kp=5 Poles: -5.65, i, i The controller design must guarantee closed loop stability

33 Stability Analitic Procedure (Try & Error) Evalute closed loop poles for every combination of the controller parameters (Kp, Td, Ti) using the model of the system. Routh-Hurwitz stability criterion A tool to evaluate if a polinomial has roots on the right-half complex plane. The method prevents form computing the whole set of roots of a higher order polinomial It can be used to evaluate stability conditions Nyquist Stability Criterion (to be studied in chapter 5)

34 Routh-Hurwitz Stability Criterion Allows to determine if there exists a root in the right-half complex plane Important: Note the notation If there exists a negative parameter, then the polinomial has at least one root in the right-half plane. Build the Routh-Hurwitz table. If there exists a negative component on the first column, then the polinomial has at least one root in the right-half plane. There are special rules to deal with degenerate cases (See Chapter 7)

35 Example Closed loop system The poles are the solution of the following equation (depends on Kp) Gains range

36 Example Closed loop system The poles are the solution of the following equation (depend on Kp y Ti) Not vey useful for multiple parameters

37 Index Closed Loop Transfer Function Tuning a Controller Stability analysis of systems Steady-state response of a closed-loop system Transient response of a stable system. Closed-loop poles and zeros vs. Controller parameters

38 Steady State Response Analysis of system response as time tends to infinity (We assume the closed loop system is stable) Steady state error Final Value Theorem (property of Laplace transform) Important: Depends on R(s) Different references define different steady-state error parameters.

39 Error for Step input Error for a constant steady-state input.8 Position error constant All stable systems have bounded steady-state errors For the error to be null (The system reaches the reference)

40 Error for Ramp input Steady state error for ramp input Velocity Error Bounded Velocity Error Null position Error (C(s)G(s) has at least one integrator) For the velocity error to be null (the system reached the reference)

41 Error for parabolic input Steady state error for parabolic input Acceleration error constant Bounded acceleration error Null position error Null velocity error (C(s)G(s) has at least two integrators) For the parabolic error to be null (The system reaches the reference)

42 Error Table Type of a system = Number of integrators Type Error Step Ramp Parabolic

43 Example Proportional Controller Type I System Proportional Controller affects the Bode Gain of the system, but can not change its Type. Improves (quantitatively) steady state behaviour. Depends on Kp.

44 Example Position error. Constant reference (Step).5 Kp =, Td =, /Ti = y(t) u(t) e(t).5 e(τ)dτ t Respuesta del sistema en BC

45 Example Velocity error. Increasing reference (ramp) 6 Kp =, Td =, /Ti = y(t) u(t) e(t) 5 e(τ)dτ t Respuesta del sistema en BC

46 Example PI Controller Type I system P controllers affect the Bode gain of the system and increases the system type Improves (qualitatively) steady state behaviour Depends on Kp and Ti (Lag controllers allow to increase Bode gain)

47 Example Position error. Constant reference (Step).5 Kp =, Td =, /Ti =. y(t) u(t) e(t).5 e(τ)dτ t Respuesta del sistema en BC

48 Example Velocity error. Increasing reference (ramp) 6 Kp =, Td =, /Ti =. The integral term is introduced to improve steady state response. y(t) u(t) (It might unstabilize the system. Ex. Try simulation with Kp=, Ti=) e(τ)dτ e(t) t Respuesta del sistema en BC

49 Index Closed Loop Transfer Function Tuning a Controller Stability analysis of systems Steady-state response of a closed-loop system Transient response of a stable system. Closed-loop poles and zeros vs. Controller parameters

50 Transient Response Response to unit step input Classification of output signal y(t) depending on the input signal. Unit Step (the most widely used). Ramp. Sinusoidal. Provides information about the dynamical properties of the system R(s) E(s) C(s) U(s) G(s) Y(s) - Response in y(t) when a reference r(t) is applied Reference signal: Unit Step signal. Shows the speed of response of the system (In general the reference signal will be different than the unit step)

51 Transient Response TEMA 5. Respuesta temporal de sistemas lineales. Sistemas dinámicos lineales de primer orden. Ejemplos. Sistemas dinámicos lineales de segundo orden. Respuesta ante escalón. Sistemas de orden n. We are interested in the output y(t) as r(t) varies in time (Closed-loop behavior) The trnasient response of a LTI system depends on the closed-loop transfer function (Gbc(s)) One option: Try & Error Given a system, simulate or aplly inverde Laplace transform It is difficult to characterize propoerties as raise time or overshoot Identify the effect of parameters in the response.

52 First Order Systems dy τ + y = Ku, y() = dt Y(s) K G(s) = = U(s) + τ s y K : Static Gain u τ : Time Constant (units according to input & output) (measured in time units) u = tiempo y y = 3.78 y = τ tiempo

53 Second order Systems d y dt + a dy dt + a y = b u d y dt + δ ω dy dt n + ωn y = K ω n u K :Static Gain (dim Y/dim U) δ : Damping Coefficient (adimensional) ω : Natural frequency ( rad/s) n Y(s) G(s) = = U(s) s Kωn + δω s+ ω n n

54 Second order Systems Poles : δ ω ± ω n n δ Im Im Re Re δ δ δ > : = : < : Overdamped Critically damped. Underdamped

55 Second order Systems Underdamped system S. O. = y( t p ) y( ) y( ) y(t) y( ) t s t p t e Tiempo t s t p = = ω ω π α n n δ π δ δ π δ..(%) = e S O t e 3 = ω n δ

56 ) ( lim gain static the K is where )] cos( ) ( [ ) ( ) ( ) ( ) ( ' ) ( G s K t c t sen b e e a K t y s s p s c s k s s Y s k k k k r k t t j t p j k k k r k j t j i m i k k j = = = = = = = = δ ω δ ω ω ω δ ω δ Higher order Systems n n n n m m m m m m m m m n n n n n n a s a a s s b bs b s G s u t b dt du t b dt u t d b dt u t d b y a dt dy t a dt t y d a dt t y d = = ) ( ) ( ) (... ) ( ) ( ) (... ) ( ) (

57 Dominant Poles In practice, some poles have more influence in the response than others. These poles are called dominant poles The dominant poles are those yielding the slowest reponse The response speed is given by the exponent of the exponential terms (the real part of the pole). Remember: Dominant dynamics: poles with the slowest response In practice, the dominat poles are determined from their relative distance to the imaginary axis. p Im p Im p is dominant if d /d >5 p d d d Re d Re p The static gain must remain the same p p p

58 Dominant Poles G( s) = 544 ( s + )( s + 6)( s + 7) 544 ( s + )(6)(7) = s + Im y(t) Re Tiempo(s) - is el dominant. The remaining poles are neglected

59 Effect of zeros in the output Zeros have an influence in the response.5 Step Response.5 Amplitude Time (sec) 6 Step Response Amplitude Time (sec)

60 Effect of zeros in the output Effect of adding a zero Qualitatively: 6 Step Response 5 4 y(t) dy(t)/dt yc(t) Amplitude Time (sec)

61 Non-minimum phase zeros o x o x o o Step Response Step Response.5 Step Response 6 Step Response Amplitude.4..8 Amplitude.4..8 Amplitude.5 Amplitude Time (sec) Time (sec) Time (sec) Time (sec)

62 Non-minimum phase zeros x x o o Step Response Step Response.5.5 Amplitude Amplitude Time (sec) Time (sec)

63 Dynamics cancellation Step Response x ox Amplitude Time (sec) The closer the zero is to the pole, the less it influences system response Affects the dominant dynamics (in transient regime) Settling time is not significantly affected.

64 Design Hypothesis Quick review of concepts Time response of first order systems Time response of second order systems Time response of higher order systems Effect of zeros It is difficult to obtain explicit results in general Design Hypothesis Explicit expressions for the effect of controller parameters on the transient response for step inputs are required. A pair of conjugate complex poles dominate the closed loop response Zeros are difficult to deal with, in general. Not considered. Two design tools: Root Locus design Frequency domain design

65 Index Closed Loop Transfer Function Tuning a Controller Stability analysis of system Steady-state response of a closed-loop systems Transient response of a stable system. Closed-loop poles and zeros vs. Controller parameters

66 Poles and Zeros of Gbc(s) Closed-loop transfer function Zeros of the closed-loop system The same zeros of the openloop plant plus those of the controller Poles of the closed-loop system Depend on the design parameter In some cases it is possible to get explicit expressions for nd order systems with P and PD controllers Not possible in general Approximate methodologies

67 Example P Controller The poles depend on Kp Complex plane representation Root Locus

68 Example.5 Kp=5 Kp=.5 Kp= Kp=

69 Illustratuve example: Magnetic levitation system Description Ball material Ball diameter Coil diameter Value Steel 5 mm 8 mm Winding turns 85 Resistence Inductance Ω 77 mh a khz 44 mh a khz

70 Illustratuve example: Magnetic levitation system Nonlinear model of the system mx&& = mg k I X X F m F g System Linealization m : Ball mass g : Gravity constant X : Distance between ball and coil (magnitudes to be controlled) I : Coil current (control action) K : constant coefficient We assume the operating point X with control action I and consider error variables I = I X = X + I + X

71 System Linearization The incremental variables depend on the selected operating point.

72 System Linearization

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