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Model Predictive Control Toolbox™ software supports the same LTI model formats as does Control System Toolbox™ software. You can use whichever is most convenient for your application. It's also easy to convert from one format to another. For more details, see the Control System Toolbox documentation.

A transfer function (TF) relates a particular input/output pair.
For example, if *u*(*t*) is a plant
input and *y*(*t*) is an output,
the transfer function relating them might be:

$$\frac{Y(s)}{U(s)}=G(s)=\frac{s+2}{{s}^{2}+s+10}{e}^{-1.5s}$$

This TF consists of a *numerator* polynomial, *s*+2,
a *denominator* polynomial, *s ^{2}*+

`tf`

function:```
Gtf1 = tf([1 2], [1 1 10],'OutputDelay',1.5)
```

Control System Toolbox software builds and displays it as follows:

Transfer function: s + 2 exp(-1.5*s) * ------------ s^2 + s + 10

Like the TF, the zero/pole/gain (ZPK) format relates an input/output pair. The difference is that the ZPK numerator and denominator polynomials are factored, as in

$$G(s)=2.5\frac{s+0.45}{(s+0.3)(s+0.1+0.7i)(s+0.1-0.7i)}$$

(zeros and/or poles are complex numbers in general).

You define the ZPK model by specifying the zero(s), pole(s), and gain as in

poles = [-0.3, -0.1+0.7*i, -0.1-0.7*i]; Gzpk1 = zpk(-0.45,poles,2.5);

The state-space format is convenient if your model is a set of LTI differential and algebraic equations. For example, consider the following linearized model of a continuous stirred-tank reactor (CSTR) involving an exothermic (heat-generating) reaction [1].

$$\frac{d{{C}^{\prime}}_{A}}{dt}={a}_{11}{{C}^{\prime}}_{A}+{a}_{12}{T}^{\prime}+{b}_{11}{{T}^{\prime}}_{c}+{b}_{12}{{C}^{\prime}}_{Ai}$$

$$\frac{d{T}^{\prime}}{dt}={a}_{21}{{C}^{\prime}}_{A}+{a}_{22}{T}^{\prime}+{b}_{21}{{T}^{\prime}}_{c}+{b}_{22}{{C}^{\prime}}_{Ai}$$

where *C _{A}* is the concentration
of a key reactant,

**CSTR Schematic**

Measurement of reactant concentrations is often difficult, if
not impossible. Let us assume that *T* is a measured
output, *C _{A}* is an unmeasured
output,

The model fits the general state-space format

$$\frac{dx}{dt}=Ax+Bu$$

$$y=Cx+Du$$

where

$$x=\left[\begin{array}{c}{{C}^{\prime}}_{A}\\ {T}^{\prime}\end{array}\right]\text{,}u=\left[\begin{array}{c}{{T}^{\prime}}_{c}\\ {{C}^{\prime}}_{Ai}\end{array}\right]\text{,}y=\left[\begin{array}{c}{T}^{\prime}\\ {{C}^{\prime}}_{A}\end{array}\right]$$

$$A=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]\text{,}B=\left[\begin{array}{cc}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}\right]\text{,}C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\text{,}D=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$$

The following code shows how to define such a model for some
specific values of the *a _{ij}* and

A = [-0.0285 -0.0014 -0.0371 -0.1476]; B = [-0.0850 0.0238 0.0802 0.4462]; C = [0 1 1 0]; D = zeros(2,2); CSTR = ss(A,B,C,D);

This defines a *continuous-time* state-space
model. If you do not specify a sampling period, a default sampling
value of zero applies. You can also specify discrete-time state-space
models. You can specify delays in both continuous-time and discrete-time
models.

In the CSTR example, the *D* matrix is zero
and the output does not instantly respond to change in the input.
The Model
Predictive Control Toolbox software prohibits direct (instantaneous)
feedthrough from a manipulated variable to an output. For example,
the CSTR model could include direct feedthrough from the unmeasured
disturbance, *C _{Ai}*, to either

The `ss`

function in the last line of the
above code creates a state-space model, `CSTR`

, which
is an *LTI object*. The `tf`

and `zpk`

commands
described in Transfer Function Models and Zero/Pole/Gain Models also
create LTI objects. Such objects contain the model parameters as well
as optional properties.

The following code sets some of the `CSTR`

model's
optional properties:

CSTR.InputName = {'T_c','C_A_i'}; CSTR.OutputName = {'T','C_A'}; CSTR.StateName = {'C_A','T'}; CSTR.InputGroup.MV = 1; CSTR.InputGroup.UD = 2; CSTR.OutputGroup.MO = 1; CSTR.OutputGroup.UO = 2; CSTR

The first three lines specify labels for the input, output and
state variables. The next four specify the signal type for each input
and output. The designations `MV`

, `UD`

, `MO`

,
and `UO`

mean *manipulated variable*, *unmeasured
disturbance*, *measured output*, and *unmeasured
output*. (See Signal Types for
definitions.) For example, the code specifies that input 2 of model `CSTR`

is
an unmeasured disturbance. The last line causes the LTI object to
be displayed, generating the following lines in the MATLAB^{®} Command
Window:

A = C_A T C_A -0.0285 -0.0014 T -0.0371 -0.1476 B = T_c C_Ai C_A -0.085 0.0238 T 0.0802 0.4462 C = C_A T T 0 1 C_A 1 0 D = T_c C_Ai T 0 0 C_A 0 0 Input groups: Name Channels MV 1 UD 2 Output groups: Name Channels MO 1 UO 2 Continuous-time model

The optional `InputName`

and `OutputName`

properties
affect the model displays, as in the above example. The software also
uses the `InputName`

and `OutputName`

properties
to label plots and tables. In that context, the underscore character
causes the next character to be displayed as a subscript.

**General Case. **As mentioned in Signal Types, Model
Predictive Control Toolbox software
supports three input types and two output types. In a Model
Predictive Control Toolbox design,
designation of the input and output types determines the controller
dimensions and has other important consequences.

For example, suppose your plant structure were as follows:

Plant Inputs | Plant Outputs |
---|---|

Two manipulated variables (MVs) | Three measured outputs (MOs) |

One measured disturbance (MD) | Two unmeasured outputs (UOs) |

Two unmeasured disturbances (UDs) |

The resulting controller has four inputs (the three MOs and the MD) and two outputs (the MVs). It includes feedforward compensation for the measured disturbance, and assumes that you wanted to include the unmeasured disturbances and outputs as part of the regulator design.

If you didn't want a particular signal to be treated as one of the above types, you could do one of the following:

Eliminate the signal before using the model in controller design.

For an output, designate it as unmeasured, then set its weight to zero.

For an input, designate it as an unmeasured disturbance, then define a custom state estimator that ignores the input.

### Note

By default, the software assumes that unspecified plant inputs are manipulated variables, and unspecified outputs are measured. Thus, if you didn't specify signal types in the above example, the controller would have four inputs (assuming all plant outputs were measured) and five outputs (assuming all plant inputs were manipulated variables).

For model `CSTR`

, the default Model
Predictive Control Toolbox assumptions
are incorrect. You must set its `InputGroup`

and `OutputGroup`

properties,
as illustrated in the above code, or modify the default settings when
you load the model into **MPC Designer**.

Use `setmpcsignals`

to make type definition.
For example:

CSTR = setmpcsignals(CSTR,'UD',2,'UO',2);

sets `InputGroup`

and `OutputGroup`

to
the same values as in the previous example. The `CSTR`

display
would then include the following lines:

Input groups: Name Channels Unmeasured 2 Manipulated 1 Output groups: Name Channels Unmeasured 2 Measured 1

Notice that `setmpcsignals`

sets unspecified
inputs to `Manipulated`

and unspecified outputs to `Measured`

.

Control System
Toolbox software provides functions for analyzing
LTI models. Some of the more commonly used are listed below. Type
the example code at the MATLAB prompt to see how they work for
the `CSTR`

example.

Example | Intended Result |
---|---|

`dcgain(CSTR)` | Calculate gain matrix for the |

`impulse(CSTR)` | Graph |

`linearSystemAnalyzer(CSTR)` | Open the Linear System Analyzer with the |

`pole(CSTR)` | Calculate |

`step(CSTR)` | Graph |

`zero(CSTR)` | Compute |

[1] Seborg, D. E., T. F. Edgar, and D. A.
Mellichamp, *Process Dynamics and Control*, 2nd
Edition, Wiley, 2004, pp. 34–36 and 94–95.

`setmpcsignals`

| `ss`

| `tf`

| `zpk`

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