This example shows how to construct a generalized state-space (`genss`

) model of a control system that has both tunable and uncertain parameters. You can use `systune`

to tune the tunable parameters of such a model to achieve performance that is robust against the uncertainty in the system.

For this example, the plant is a mass-spring-damper system. The input is the applied force, *F*, and the output is *x*, the position of the mass.

In this system, the mass *m*, the damping constant *c*, and the spring constant *k* all have some uncertainty. Use uncertain `ureal`

parameters to represent these quantities in terms of their nominal or most probable value and a range of uncertainty around that value.

um = ureal('m',3,'Percentage',40); uc = ureal('c',1,'Percentage',20); uk = ureal('k',2,'Percentage',30);

The transfer function of a mass-spring-damper system is a second-order function given by:

$$G\left(s\right)=\frac{1}{m{s}^{2}+cs+k}.$$

Create this transfer function in MATLAB® using the uncertain parameters and the `tf`

command. The result is an uncertain state-space (`uss`

) model.

G = tf(1,[um uc uk])

G = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 2 states. The model uncertainty consists of the following blocks: c: Uncertain real, nominal = 1, variability = [-20,20]%, 1 occurrences k: Uncertain real, nominal = 2, variability = [-30,30]%, 1 occurrences m: Uncertain real, nominal = 3, variability = [-40,40]%, 1 occurrences Type "G.NominalValue" to see the nominal value, "get(G)" to see all properties, and "G.Uncertainty" to interact with the uncertain elements.

Suppose you want to control this system with a PID controller, and that your design requirements include monitoring the response to noise at the plant input. Build a model of the following control system.

Use a tunable PID controller, and insert an analysis point to provide access to the disturbance input.

C0 = tunablePID('C','PID'); d = AnalysisPoint('d');

Connect all the components to create the control system model.

T0 = feedback(G*d*C0,1)

T0 = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 3 states, and the following blocks: C: Parametric PID controller, 1 occurrences. c: Uncertain real, nominal = 1, variability = [-20,20]%, 1 occurrences d: Analysis point, 1 channels, 1 occurrences. k: Uncertain real, nominal = 2, variability = [-30,30]%, 1 occurrences m: Uncertain real, nominal = 3, variability = [-40,40]%, 1 occurrences Type "ss(T0)" to see the current value, "get(T0)" to see all properties, and "T0.Blocks" to interact with the blocks.

T0.InputName = 'r'; T0.OutputName = 'x';

`T0`

is a generalized state-space (`genss`

) model that has both tunable and uncertain blocks. In general, you can use `feedback`

and other model interconnection commands, such as `connect`

, to build up models of more complex tunable and uncertain control systems from fixed-value LTI components, uncertain components, and tunable components.

When you plot system responses of a `genss`

model that is both tunable and uncertain, the plot displays multiple responses computed at random values of the uncertain components. This sampling provides a general sense of the range of possible responses. All plots use the current value of the tunable components.

bodeplot(T0)

When you extract responses from a tunable and uncertain `genss`

model, the responses also contain both tunable and uncertain blocks. For example, examine the loop transfer function at the disturbance input.

`S0 = getLoopTransfer(T0,'d')`

S0 = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 3 states, and the following blocks: C: Parametric PID controller, 1 occurrences. c: Uncertain real, nominal = 1, variability = [-20,20]%, 1 occurrences d: Analysis point, 1 channels, 1 occurrences. k: Uncertain real, nominal = 2, variability = [-30,30]%, 1 occurrences m: Uncertain real, nominal = 3, variability = [-40,40]%, 1 occurrences Type "ss(S0)" to see the current value, "get(S0)" to see all properties, and "S0.Blocks" to interact with the blocks.

bodeplot(S0)

You can now create tuning goals and use `systune`

to tune the PID controller coefficients of T0. When you do so, `systune`

automatically tunes the coefficients to maximize performance over the full range of uncertainty.

`AnalysisPoint`

| `connect`

| `genss`

| `ureal`