Linearization of Multirate Models
This example shows the process that the command linearize uses when extracting a linear model of a nonlinear multirate Simulink model. To illustrate the concepts, the process is first performed using functions from the Control System Toolbox before it is repeated using the linearize command.
In the Simulink model scdmrate.slx there are three different sample rates specified in five blocks. These blocks are:
- sysC - a continuous linear block,
- Integrator - a continuous integrator,
- sysTs1 - a block that has a sample time of 0.01 seconds,
- sysTs2 - a block that has a sample time of 0.025 seconds, and
- Zero-Order Hold - a block that samples the incoming signal at 0.01 seconds.
sysC = zpk(-2,-10,0.1); Integrator = zpk(,0,1); sysTs1 = zpk(-0.7463,[0.4251 0.9735],0.2212,0.01); sysTs2 = zpk(,0.7788,0.2212,0.025);
The model below shows how the blocks are connected.
In this example we linearize the model between the output of the Constant block and the output of the block sysTs2.
Step 1: Linearizing the Blocks in the Model
The first step of the linearization is to linearize each block in the model. The linearization of the Saturation and Zero-Order Hold blocks is 1. The LTI blocks are already linear and therefore remain the same. The new model with linearized blocks is shown below.
Step 2: Rate Conversions
Because the blocks in the model contain different sample rates, it is not possible to create a single-rate linearized model for the system without first using rate conversion functions to convert the various sample rates to a representative single rate. The rate conversion functions use an iterative method. The iterations begin with a least common multiple of the sample times in the model. In this example the sample times are 0, 0.01, and 0.025 seconds which yields a least common multiple of 0.05. The rate conversion functions then take the combination of blocks with the fastest sample rate and resample them at the next fastest sample rate. In this example the first iteration converts the combination of the linearized continuous time blocks, sysC and integrator to a sample time of 0.01 using a zero order hold continuous to discrete conversion.
sysC_Ts1 = c2d(sysC*Integrator,0.01);
The blocks sysC and Integrator are now replaced by sysC_Ts1.
The next iteration converts all the blocks with a sample time of 0.01 to a sample time of 0.025. First, the following command represents the combination of these blocks by closing the feedback loop.
sysCL = feedback(sysTs1*sysC_Ts1,1);
Next, a zero-order hold method converts the closed loop system, sysCL, from a sample rate of 0.01 to 0.025.
sysCL_Ts2 = d2d(sysCL,0.025);
The system sysCL_Ts2 then replaces the feedback loop in the model.
The final iteration re-samples the combination of the closed loop system and the block sysTs2 from a rate of 0.025 seconds to a rate of 0.05 seconds.
sys_L = d2d(sysCL_Ts2*sysTs2,0.05)
sys_L = 0.0001057 (z+22.76) (z+0.912) (z-0.9048) (z+0.06495) ------------------------------------------------------- (z-0.01373) (z-0.6065) (z-0.6386) (z-0.8588) (z-0.9754) Sample time: 0.05 seconds Discrete-time zero/pole/gain model.
Linearizing the Model Using Simulink Control Design Commands
We can reproduce these results using the command line interface of Simulink Control Design.
model = 'scdmrate'; io(1) = linio('scdmrate/Constant',1,'input'); io(2) = linio('scdmrate/sysTs2',1,'openoutput'); sys = zpk(linearize(model,io))
sys = From input "Constant" to output "sysTs2": 0.0001057 (z+22.76) (z+0.912) (z-0.9048) (z+0.06495) ------------------------------------------------------- (z-0.6065) (z-0.6386) (z-0.8588) (z-0.9754) (z-0.01373) Sample time: 0.05 seconds Discrete-time zero/pole/gain model.