Exercise 11: Discretization

Transcription

1 Exercise 11: Discretization Introduction We will use different discretization methods using pen and paper exercises and in practical implementation in MathScript/LabVIEW along with built-in discretization methods. Euler: Two commonly used methods for discretization are as follows: Euler forward method: Euler backward method: Other methods are Zero Order Hold (ZOH), Tustin s method, etc. Where is the sampling time, and, and are discrete values. Note! Different notations may be used: =,,, etc. Example: Given the following system: We will use the Euler forward method to create a discrete version of this system. We get as follows: Further: Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: Fax:

2 2 This gives: Finally: If we use, and, we get: [End of Example] Discretization in MathScript In MathScript we can simulate a discrete system using a For Loop or a While Loop. We can also use some of the built-in functions to convert a continuous state-space model to a discrete state-space model. In MathScript we can use the function c2d() to convert from a continuous system to a discrete system. Example: We will simulate the following discrete system in MathScript using a For Loop: We will set and MathScript code: % Simulation of discrete model clear clc % Model Parameters a = 0.25; b = 2; % Simulation Parameters Ts = 0.1; %s Tstop = 20; %s uk = 1; x(1) = 0; % Simulation for k=1:(tstop/ts) x(k+1) = (1-a*Ts).*x(k) + Ts*b*uk; end % Plot the Simulation Results k=0:ts:tstop;

3 3 plot (k, x) grid on We use as the sampling time. This gives the following results: As you may notice, we get the same result as we get in the previous Exercise (10b) when we used LabVIEW. [End of Example] Example: Given the following system: We will set and We will use the built-in function c2d() in order to find the discrete system: % Find Discrete model clear clc % Model Parameters a = 0.25; b = 2; Ts = 0.1; %s % State-space model A = [-a];

4 4 B = [b]; C = [1]; D = [0]; model = ss(a,b,c,d) model_discrete = c2d(model, Ts, 'forward') This gives the following results: a b 0.2 c 1 d 0 or: As you see this is the same result as in the example above. [End of Example] Discretization in LabVIEW There are several ways we can make a discrete model in LabVIEW as well. In this exercise we will use the Formula Node. A Formula Node in LabVIEW evaluates mathematical formulas and expressions similar to C on the block diagram. In this way you may use existing C code directly inside your LabVIEW code. It is also useful when you have complex mathematical expressions. Example: We use the same discrete system as in previous examples: With and We will use the Formula Node and a For Loop in LabVIEW in order to simulate this discrete model. The LabVIEW code is as follows:

5 5 The simulation results are as follows: As you see we get the same results here as in the other examples. [End of Example] Task 1: Discrete system Given the following system:

6 6 Task 1.1 Find the discrete system and set it on state-space form (using pen and paper). Use Euler forward: Task 1.2 Define the continuous state-space model ( pen and paper ) and then implement the continuous state-space model in MathScript. Then use MathScript to find the discrete state-space model. Compare the result from the previous task. Set and Task 2: Discrete Controller A controller is given by the following transfer function: Task 2.1 Find the continuous differential equation. Task 2.2 Find the discrete difference equation. Use the Euler backward method. Task 3: Discrete State-space model Given the following system: Task 3.1 Find the discrete state-space model

7 7 Task 3.2 Define the continuous state-space model ( pen and paper ) and then implement the continuous state-space model in MathScript. Then use MathScript to find the discrete state-space model. Compare the result from the previous subtask. Use values and. Task 4: Discrete Low-pass Filter Transfer function for a first-order low-pass filter may be written: Where is the time-constant of the filter. Task 4.1 Create the discrete low-pass filter algorithm using pen and paper. Use the Euler Backward method. Task 4.2 Create a discrete low-pass filter in LabVIEW using the Formula Node in LabVIEW. Create a SubVI of the code. You will use this subvi in your project later. The user needs to be able to set the time constant of the filter from the outside, i.e., it should be an input to the SubVI. The simulation Time-step Test and make sure your filter works! Note! A golden rule is that: needs also to be set from the outside. You may, e.g., use the Uniform White Noise PtByPt.vi. Example:

8 8 Task 5: Discrete PI controller A continuous-time PI controller may be written: Where u is the controller output and is the control error: Below we see a block diagram of a simple control system:

9 9 Task 5.1 Create the discrete PI Controller algorithm using pen and paper. Use the Euler Backward method. Task 5.2 Create a discrete PI controller in LabVIEW using the Formula Node. Create a SubVI of the code. You will use this subvi in your project later. Task 6: Simulation Implement your PI controller and Low-pass filter in a simulation. You may, e.g., use the example General PID Simulator.vi as a base for your simulation. Use the NI Example Finder (Help Find Examples ) in order to find the VI in LabVIEW. Note! You will need the LabVIEW PID and Fuzzy Logic Toolkit in order to fulfill this Task. Run the example and see how it is implemented and how it works. Note! Save the VI with a new name and replace the controller used in the example with the controller you created in the previous task. Additional Resources Here you will find tutorials, additional exercises, etc.