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Advanced DYNAMICS and CONTROL ... 13 Stability analysis of discrete-time dynamic systems ... Analysis and design of advanced controllers and estimators:.
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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
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
[ (^) x 1 x 2
︸ ︷︷ x ︸
^ d^11 d^12 d 21 d 22
[ (^) 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 ˙ ︸
− K mf − (^) mD
[ (^) x 1 x 2
︸ ︷︷ x ︸
m
u (1.21)
y = [︸^1 ︷︷ 0 ︸] C
[ (^) x 1 x 2
︸ ︷︷ x ︸
D
u (1.22)
[End of Example 1.2]
1.4 Linearization of non-linear models
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:
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).