Uncertain State-Space Models

Uncertain state-space (uss) models are linear systems with uncertain state-space matrices and/or uncertain linear dynamics. Like their numeric (i.e., not uncertain) counterpart, the ss model object, you can build them from state-space matrices using the ss command. When one or more of the state-space matrices contain uncertain elements (uncertain Control Design Blocks), the result is a uss model object.

Combining uncertain systems with other uncertain systems (for example, using model arithmetic, connect, or feedback) usually results in an uncertain system. You can also combine numeric systems with uncertain systems. Usually the result is an uncertain system. The nominal value of an uncertain system is an ss model object.

In the example below, the A, B and C matrices are made up of uncertain real parameters. Packing them together with the ss command results in a continuous-time uncertain system.

Uncertain State-Space Model

To create an uncertain state-space model, you first use Control Design Blocks to create uncertain elements. Then, use the elements to specify the state-space matrices of the system.

For instance, create three uncertain real parameters and build state-spaces matrices from them.

p1 = ureal('p1',10,'Percentage',50); 
p2 = ureal('p2',3,'PlusMinus',[-.5 1.2]); 
p3 = ureal('p3',0); 

A = [-p1 p2; 0 -p1]; 
B = [-p2; p2+p3]; 
C = [1 0; 1 1-p3]; 
D = [0; 0];

The matrices constructed with uncertain parameters, A, B, and C, are uncertain matrix (umat) objects. Using them as inputs to ss results in a 2-output, 1-input, 2-state uncertain system.

sys = ss(A,B,C,D)

Uncertain continuous-time state-space model with 2 outputs, 1 inputs, 2 states.
The model uncertainty consists of the following blocks:
  p1: Uncertain real, nominal = 10, variability = [-50,50]%, 2 occurrences
  p2: Uncertain real, nominal = 3, variability = [-0.5,1.2], 2 occurrences
  p3: Uncertain real, nominal = 0, variability = [-1,1], 2 occurrences

Type "sys.NominalValue" to see the nominal value and "sys.Uncertainty" to interact with the uncertain elements.

The display shows that the system includes the three uncertain parameters.

Properties of uss Objects

uss models, like all model objects, include properties that store dynamics and model metadata. View the properties of an uncertain state-space model.

p1 = ureal('p1',10,'Percentage',50);
p2 = ureal('p2',3,'PlusMinus',[-.5 1.2]);
p3 = ureal('p3',0);
A = [-p1 p2; 0 -p1];
B = [-p2; p2+p3];
C = [1 0; 1 1-p3];
D = [0; 0];
sys = ss(A,B,C,D);     % create uss model

get(sys)

     NominalValue: [2x1 ss]
      Uncertainty: [1x1 struct]
                A: [2x2 umat]
                B: [2x1 umat]
                C: [2x2 umat]
                D: [2x1 double]
                E: []
        StateName: {2x1 cell}
        StateUnit: {2x1 cell}
    InternalDelay: [0x1 double]
       InputDelay: 0
      OutputDelay: [2x1 double]
        InputName: {''}
        InputUnit: {''}
       InputGroup: [1x1 struct]
       OutputName: {2x1 cell}
       OutputUnit: {2x1 cell}
      OutputGroup: [1x1 struct]
            Notes: [0x1 string]
         UserData: []
             Name: ''
               Ts: 0
         TimeUnit: 'seconds'
     SamplingGrid: [1x1 struct]

Most of the properties behave similarly to how they behave for ss model objects. The NominalValue property is itself an ss model object. You can therefore analyze the nominal value as you would any state-space model. For instance, compute the poles and step response of the nominal system.

pole(sys.NominalValue)

ans = 2×1

   -10
   -10

step(sys.NominalValue)

As with the uncertain matrices (umat), the Uncertainty property is a structure containing the uncertain elements. You can use this property for direct access to the uncertain elements. For instance, check the Range of the uncertain element named p2 within sys.

sys.Uncertainty.p2.Range

ans = 1×2

    2.5000    4.2000

Change the uncertainty range of p2 within sys.

sys.Uncertainty.p2.Range = [2 4];

This command changes only the range of the parameter called p2 in sys. It does not change the variable p2 in the MATLAB® workspace.

p2.Range

ans = 1×2

    2.5000    4.2000

Lifting an ss to a uss

A not-uncertain state space object may be interpreted as an uncertain state space object that has no dependence on uncertain elements. Use the uss command to “lift” an ss to the uss class.

sys = rss(3,2,1); 
usys = uss(sys) 
USS: 3 States, 2 Outputs, 1 Input, Continuous System 

Arrays of ss objects can also be lifted. See Array Management for Uncertain Objects for more information about how arrays of uncertain objects are handled.