Simulation

Simulation means computing the model response using input data and initial conditions. The time samples of the model response match the time samples of the input data used for simulation. In other words, given inputs u(t₁, … ,t_N), the simulation generates y(t₁, … ,t_N). The following diagram illustrates this flow.

For a continuous-time system, simulation means solving a differential equation. For a discrete-time system, simulation means directly applying the model equations.

For example, consider a dynamic model described by a first-order difference equation that uses a sample time of 1 second:

y(t+1) = ay(t) + bu(t),

where y is the output and u is the input.

This system is equivalent to the following block diagram.

Suppose that your model identification provides you with estimated parameter values of a = 0.9 and b = 1.5. Then the equation becomes:

y(t+1) = 0.9y(t) + 1.5u(t).

Now suppose that you want to compute the values y(1), y(2), y(3),... for given input values u(0) = 2, u(1) = 1, u(2) = 4,… Here, y(1) is the value of output at the first sampling instant. Using initial condition of y(0) = 0, the values of y(t) for times t = 1, 2, and 3 can be computed as:

y(1) = 0.9y(0) + 1.5u(0) =(0.9)(0) + (1.5)(2) = 3

y(2) = 0.9y(1) + 1.5u(1) =(0.9)(3) + (1.5)(1) = 4.2

y(3) = 0.9y(2) + 1.5u(2) =(0.9)(4.2) + (1.5)(4) = 9.78

Prediction

Prediction means projecting the model response k steps ahead into the future using the current and past values of measured input and output values. k is called the prediction horizon, and corresponds to predicting output at time kT_s, where T_s is the sample time. In other words, given measured inputs u_m(t₁, … ,t_N+k) and measured outputs y_m(t₁, … ,t_N) , the prediction generates y_p(t_N+k).

For example, suppose that you use sensors to measure the input signal u_m(t) and output signal y_m(t) of the physical system described in the previous first-order equation. The equation becomes:

y_p(t+1) = ay_m(t) + bu_m(t),

where y is the output and u is the input.

The predictor version of the previous simulation block diagram is:

At the 10th sampling instant (t = 10), the measured output y_m(10) is 16 mm and the corresponding input u_m(10) is 12 N. Now, you want to predict the value of the output at the future time t = 11. Using the previous equation, the predicted output y_p is:

y_p(11) = 0.9y_m(10) + 1.5u_m(10)

Hence, the predicted value of future output y(11) at time t = 10 is:

y_p(11) = 0.9*16 + 1.5*12 = 32.4

In general, to predict the model response k steps into the future (k≥1) from the current time t, you must know the inputs up to time t+k and outputs up to time t:

y_p(t+k) = f(u_m(t+k),u_m(t+k–1),...,u_m(t),u_m(t–1),...,u_m(0),
                y_m(t),y_m(t–1),y_m(t–2),...,y_m(0))

u_m(0) and y_m(0) are the initial states. f() represents the predictor, which is a dynamic model whose form depends on the model structure.

The one-step-ahead predictor from the previous example, y_p of the model y(t) + ay(t–1) = bu(t) is:

y_p(t+1) = –ay_p(t) + bu_m(t+1)

In this simple one-step predictor case, the newest prediction is based only on measurements. For multiple-step predictors, the dynamic model propagates states internally, using the previous predicted states in addition to the inputs. Each predicted output therefore arises from a combination of the measured input and outputs and the previous predicted outputs.

You can set k to any positive integer value up to the number of measured data samples. If you set k to ∞, then no previous outputs are used in the prediction computation, and prediction returns the same result as simulation. If you set k to an integer greater than the number of data samples, predict sets k to Inf and issues a warning. If your intent is to perform a prediction in a time range beyond the last instant of measured data, use forecast.

For an example showing prediction and simulation in MATLAB^®, see Compare Predicted and Simulated Response of Identified Model to Measured Data.

Limitations on Prediction

Not all models support a predictive approach. For the previous dynamic model, $$H(z)=\frac{1}{1+a{z}^{-1}}$$, the structure supports use of past data. This support does not exist in models of Output-Error (OE) structure (H(z) = 1). There is no information in past outputs that can be used for predicting future output values. In these cases, prediction and simulation coincide. Even models that generally do use past information can still have OE structure in special cases. State-space models (idss) have OE structure when K=0. Polynomial models (idpoly) also have OE structure, when a=c=d=1. In these special cases, prediction and simulation are equivalent, and the disturbance model is fixed to 1.