Advanced DYNAMICS and CONTROL, Lecture notes of Dynamics

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Advanced DYNAMICS and CONTROL

Finn Haugen

TechTeach

August 2010

ISBN 978-82-91748-17-

• I CONTINUOUS-TIME SYSTEMS THEORY
• 1 State-space models
  • 1.1 Introduction
  • 1.2 A general state-space model
  • 1.3 Linear state-space models
  • 1.4 Linearization of non-linear models
    • 1.4.1 Introduction
    • 1.4.2 Deriving the linearization formulas
• 2 Frequency response
  • 2.1 Introduction
    • and output 2.2 How to calculate frequency response from sinusoidal input
  • 2.3 How to calculate frequency response from transfer functions
  • 2.4 Application of frequency response: Signal filters
    • 2.4.1 Introduction
    • 2.4.2 First order lowpass filters
• 3 Frequency response analysis of feedback control systems
  • 3.1 Introduction
  • 3.2 Definition of setpoint tracking and disturbance compensation
  • 3.3 Definition of characteristic transfer functions
    • 3.3.1 The Sensitivity transfer function
    • 3.3.2 The Tracking transfer function
    • bance compensation 3.4 Frequency response analysis of setpoint tracking and distur-
    • 3.4.1 Introduction
    • 3.4.2 Frequency response analysis of setpoint tracking
      • sation 3.4.3 Frequency response analysis of disturbance compen-
• 4 Stability analysis of dynamic systems
  • 4.1 Introduction
  • 4.2 Stability properties and impulse response
  • 4.3 Stability properties and poles
  • 4.4 Stability properties of state-space models
• 5 Stability analysis of feedback systems
  • 5.1 Introduction
  • 5.2 Pole-based stability analysis of feedback systems
  • 5.3 Nyquist’s stability criterion
  • 5.4 Stability margins - margin 5.4.1 Stability margins in terms of gain margin and phase - plitude 5.4.2 Stability margins in terms of maximum sensitivity am-
  • 5.5 Stability analysis in a Bode diagram
  • 5.6 Robustness in term of stability margins
• II DISCRETE-TIME SYSTEMS THEORY
• 6 Discrete-time signals
• 7 Difference equations
  • 7.1 Difference equation models
  • 7.2 Calculating responses from difference equation models
• 8 Discretizing continuous-time models
  • 8.1 Simple discretization methods
  • 8.2 Discretizing a simulator of a dynamic system
  • 8.3 Discretizing a signal filter
  • 8.4 Discretizing a PID controller
    • 8.4.1 Computer based control loop
    • 8.4.2 Development of discrete-time PID controller
    • 8.4.3 Some practical features of the PID controller
    • 8.4.4 Selecting the sampling time of the control system
• 9 Discrete-time state space models
  • 9.1 General form of discrete-time state space models
  • 9.2 Linear discrete-time state space models
  • 9.3 Discretization of continuous-time state space models - models 9.3.1 Discretization of non-linear continuous-time state-space
    • els 9.3.2 Discretization of linear continuous-time state-space mod-
• 10 The z-transform
  • 10.1 Definition of the z-transform
  • 10.2 Properties of the z-transform
  • 10.3 z-transform pairs
  • 10.4 Inverse z-transform
• 11 Discrete-time (or z-) transfer functions
  • 11.1 Introduction
  • 11.2 From difference equation to transfer function
  • 11.3 From transfer function to difference equation
  • 11.4 Calculating time responses for discrete-time transfer functions
  • 11.5 Static transfer function and static response
  • 11.6 Poles and zeros
  • 11.7 From s-transfer functions to z-transfer functions
• 12 Frequency response of discrete-time systems
• 13 Stability analysis of discrete-time dynamic systems
  • 13.1 Definition of stability properties
  • 13.2 Stability analysis of transfer function models
  • 13.3 Stability analysis of state space models
• 14 Analysis of discrete-time feedback systems
• III STOCHASTIC SIGNALS
• 15 Stochastic signals
  • 15.1 Introduction
  • 15.2 How to characterize stochastic signals
    • 15.2.1 Realizations of stochastic processes
    • 15.2.2 Probability distribution of a stochastic variable
    • 15.2.3 The expectation value and the mean value
    • 15.2.4 Variance. Standard deviation
    • 15.2.5 Auto-covariance. Cross-covariance
  • 15.3 White and coloured noise
    • 15.3.1 White noise
    • 15.3.2 Coloured noise
    • systems 15.4 Propagation of mean value and co-variance through static
• IV ESTIMATION OF PARAMETERS AND STATES
• 16 Estimation of model parameters
  • 16.1 Introduction
    • (LS) method 16.2 Parameter estimation of static models with the Least squares
    • 16.2.1 The standard regression model
    • 16.2.2 The LS problem
    • 16.2.3 The LS solution
      • value 16.2.4 Criterion for convergence of estimate towards the true
    • 16.2.5 How to compare and select among several candidates?
  • 16.3 Parameter estimation of dynamic models
    • 16.3.1 Introduction
    • 16.3.2 Good excitation is necessary!
    • 16.3.3 How to check that a model is good?
      • LS-method 16.3.4 Estimation of differential equation models using the
    • 16.3.5 Estimation of black-box models using subspace methods
• 17 State estimation with observers
  • 17.1 Introduction
  • 17.2 How the observer works
  • 17.3 How to design observers
    • 17.3.1 Deriving the estimation error model
    • 17.3.2 Calculation of the observer gain
  • 17.4 Observability test of continuous-time systems
  • 17.5 Discrete-time implementation of the observer
  • 17.6 Estimating parameters and disturbances with observers
  • 17.7 Using observer estimates in controllers
    • sensor failure 17.8 Using observer for increased robustness of feedback control at
• 18 State estimation with Kalman Filter
  • 18.1 Introduction
  • 18.2 Observability of discrete-time systems
  • 18.3 The Kalman Filter algorithm
    • 18.3.1 The basic Kalman Filter algorithm
    • 18.3.2 Practical issues
    • 18.3.3 Features of the Kalman Filter
  • 18.4 Tuning the Kalman Filter
  • 18.5 Estimating parameters and disturbances with Kalman Filter
  • 18.6 Using the Kalman Filter estimates in controllers
    • control at sensor failure 18.7 Using the Kalman Filter for increased robustness of feedback
• V MODEL-BASED CONTROL
  • ulators 19 Testing robustness of model-based control systems with sim-
• 20 Feedback linearization
  • 20.1 Introduction
  • 20.2 Deriving the control function
    • 20.2.1 Case 1: All state variables are controlled
    • 20.2.2 Case 2: Not all state variables are controlled
• 21 LQ (Linear Quadratic) optimal control
  • 21.1 Introduction
  • 21.2 The basic LQ controller
  • 21.3 LQ controller with integral action
    • 21.3.1 Introduction
    • 21.3.2 Including integrators in the controller
    • 21.3.3 Discrete-time implementation of the LQ controller
• 22 Model-based predictive control (MPC)
• 23 Dead-time compensator (Smith predictor)
• A Model-based PID tuning with Skogestad’s method
  • A.1 The principle of Skogestad’s method
  • A.2 The tuning formulas in Skogestad’s method
  • A.3 How to find model parameters from experiments
  • A.4 Transformation from serial to parallel PID settings
  • A.5 When the process has no time-delay

Preface

This book covers estimation of model parameters and states and model-based control, and the systems theory required. I have selected the topics among many possible topics by considering what I assume are the most relevant topics in an application oriented course about these topics. It is assumed that you — the reader — has basic knowledge about complex number, differential equations, and Laplace transform based transfer functions (s-transfer functions), and also that you have knowledge about basic control methods, i.e. PID control, feedforward control and cascade control. A reference for these topics is my book Basic Dynamics and Control [5]. Supplementary material, as tutorials, ready-to-run simulators, and instructional videos are available at techteach.no. This book is available for sale only from the web site http://techteach.no. It is not allowed to make copies of the book. If you want to know about my background, please visit my home page http://techteach.no/adm/fh.

Finn Haugen, MSc TechTeach Skien, Norway, August 2010

Part I

CONTINUOUS-TIME

SYSTEMS THEORY

the left side. A general n’th order state-space model is x ˙ 1 = f 1 () (1.1) ... x ˙n = fn() (1.2) where f 1 (),... , fn() are functions (given by the model equations, of course). The variables which have their time-derivatives in the state-space model are the state-variables of the model.^1 Thus, in the model above the x-variables are state-variables. x is a common name for state-variable, but you can use any name. The initial state (at t = 0) are defined by the values x 1 (0),... , xn(0). Some times you want to define the output variables of a state-space model. y is a common name for output variable. A model with m output variables can be written y 1 = g 1 () (1.3) ... ym = gm() (1.4) where g 1 (),... , gm() are functions. We can use the name state-space model for (1.1) — (1.2) and for (1.1) — (1.4). The following example shows an example of a state-space model. A second order differential equation (which here is the model of a mass-spring-damper-system) will be written as a second order state-space model. We do this by defining a variable for each of the variables which have their time-derivative in the differential equation. These new variables becomes the state-variables of the state-space model. The same principle is used also for developing state-space models from higher order differential equations.

Example 1.1 Mass-spring-damper-model written as a state-space model

Figure 1.1 shows a mass-spring-damper-system.z is position. F is applied force. D is damping constant. K is spring constant. It is assumed that the damping force Fd is proportional to the velocity, and that the spring force (^1) The name state-space model is because the values of the state-variables x 1 (t),... , xn(t) defines thethe state-space state. of the at any instant of time. These values can be regarded as points in

m

K [N/m]

D [N/(m/s)]

F [N]

0 z [m]

Figure 1.1: Mass-spring-damper

Fs is proportional to the position of the mass. The spring force is assumed to be zero when y is zero. Force balance (Newtons 2. Law) yields my¨(t) = F (t) − Fd(t) − Fs(t) = F (t) − D y˙(t) − Ky(t) (1.5) which is a second order differential equation. We define the following new variables: x 1 for position z, x 2 for speed z˙ and u for force F. Then the model (1.5) can be written as the following equivalent set of two first order differential equations: x ˙ 1 = x 2 (1.6) m x˙ 2 = −Dx 2 − Kf x 1 + u (1.7)

which can be written on the standard form (1.1), (1.2): x ˙ 1 = (^) ︸︷︷︸x 2 f 1

x ˙ 2 = (^) ︸ m^1 (−Dx 2 −︷︷ K f x 1 + u︸) f 2

Let us regard the position x 1 as the output variable y: y = (^) ︸︷︷︸x 1 g

The initial position, x 1 (0), and the initial speed, x 2 (0), define the initial state of the system. (1.8) and (1.9) and (1.10) constitute a second order state-space model which is equivalent to the original second order differential equation (1.5). [End of Example 1.1]

which can be written on matrix-vector form as follows: [ (^) y 1 y 2

]

︸ ︷︷ y ︸

^ c^11 c^12 c 21 c 22

︸ ︷︷ C ︸

[ (^) x 1 x 2

]

︸ ︷︷ x ︸

^ d^11 d^12 d 21 d 22

︸ ︷︷ D ︸

[ (^) u 1 u 2

]

︸ ︷︷ u ︸

or, more compact: y = Cx + Du (1.20)

Example 1.2 Mass-spring-damper model written on state-space form

The state-space model (1.8), (1.9), (1.10) is linear. We get [ (^) x˙ 1 x ˙ 2

]

︸ ︷︷x ˙ ︸

^0

− K mf − (^) mD

︸ ︷︷ A ︸

[ (^) x 1 x 2

]

︸ ︷︷ x ︸

^01

m

︸ ︷︷ B ︸

u (1.21)

y = [︸^1 ︷︷ 0 ︸] C

[ (^) x 1 x 2

]

︸ ︷︷ x ︸

+ [︸^ ︷︷ 0 ︸]

D

u (1.22)

[End of Example 1.2]

1.4 Linearization of non-linear models

1.4.1 Introduction

In many cases the mathematical model contains one or more non-linear differential equations. If the mathematical model is non-linear, there may be good reasons to linearize it, which means to develop a local linear model which approximates the original model about a given operating point. The reasons may be the following:

• We want to study the behavior of the system about an operating point, which is one specified state where the system can be. It is then the deviations from this operating point we study. Examples of such operating points are the level of 8.7 m in a tank, the temperature of 50 degrees Celcius in a heat exchanger, etc. It can be shown (and we will do it soon) that a model which describes the behavior of the deviations about the operating point, is approximately linear.

• We can use the large number of the methods which are available for analysis and design of linear systems, e.g. for stability analysis, frequency response, controller design and signal filter design. The number of methods for linear models are much larger than for non-linear models.s

Note: If you have a non-linear model of a (physical) system, do not use the linearized model for simulation unless you have a good reason for using it. In stead, use the (original) non-linear model since it gives a more accurate representation of the system. Figure 1.2 illustrates the relation between the original non-linear system and the local linear system (model). The input variable which excites the

Figure 1.2: Illustration of the relation between the original non-linear system and the local linear system (model) non-linear system is assumed to be given by u = u 0 + ∆u (1.23) where u 0 is the value in the operating point and ∆u is the deviation from u 0. Similarly, x = x 0 + ∆x (1.24)

If you are going to experiment with the system to develop or adjust a linear model about the operating point, you must adjust ∆u and observe the corresponding response in ∆x (or in the output variable ∆y).