Dealing with and understanding the effects of uncertainty are important tasks for the control engineer. Reducing the effects of some forms of uncertainty (initial conditions, low-frequency disturbances) without catastrophically increasing the effects of other dominant forms (sensor noise, model uncertainty) is the primary job of the feedback control system.
Closed-loop stability is the way to deal with the (always present) uncertainty in initial conditions or arbitrarily small disturbances.
High-gain feedback in low-frequency ranges is a way to deal with the effects of unknown biases and disturbances acting on the process output. In this case, you are forced to use roll-off filters in high-frequency ranges to deal with high-frequency sensor noise in a feedback system.
Finally, notions such as gain and phase margins (and their generalizations) help quantify the sensitivity of stability and performance in the face of model uncertainty, which is the imprecise knowledge of how the control input directly affects the feedback variables.
Robust Control Toolbox™ software has built-in features allowing you to specify model uncertainty simply and naturally. The primary building blocks, called uncertain elements (or uncertain Control Design Blocks) are uncertain real parameters and uncertain linear, time-invariant objects. These can be used to create coarse and simple or detailed and complex descriptions of the model uncertainty present within your process models.
Once formulated, high-level system robustness tools can help you analyze the potential degradation of stability and performance of the closed-loop system brought on by the system model uncertainty.
The two dominant forms of model uncertainty are as follows:
Uncertainty in parameters of the underlying differential equation models
Frequency-domain uncertainty, which often quantifies model uncertainty by describing absolute or relative uncertainty in the process's frequency response
Using these two basic building blocks, along with conventional system creation commands (such as ss and tf), you can easily create uncertain system models.
An uncertain parameter has a name (used to identify it within an uncertain system with many uncertain parameters) and a nominal value. Being uncertain, it also has variability, described in one of the following ways:
An additive deviation from the nominal
A range about the nominal
A percentage deviation from the nominal
Create a real parameter, with name '|bw|', nominal value 5, and a percentage uncertainty of 10%.
bw = ureal('bw',5,'Percentage',10)
bw = Uncertain real parameter "bw" with nominal value 5 and variability [-10,10]%.
This command creates a ureal object that stores a number of parameters in its properties. View the properties of bw.
Name: 'bw' NominalValue: 5 Mode: 'Percentage' Range: [4.5000 5.5000] PlusMinus: [-0.5000 0.5000] Percentage: [-10 10] AutoSimplify: 'basic'
Note that the range of variation (Range property) and the additive deviation from nominal (the PlusMinus property) are consistent with the Percentage property value.
You can create state-space and transfer function models with uncertain real coefficients using ureal objects. The result is an uncertain state-space (uss) object. As an example, use the uncertain real parameter bw to model a first-order system whose bandwidth is between 4.5 and 5.5 rad/s.
H = tf(1,[1/bw 1])
H = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 1 states. The model uncertainty consists of the following blocks: bw: Uncertain real, nominal = 5, variability = [-10,10]%, 1 occurrences Type "H.NominalValue" to see the nominal value, "get(H)" to see all properties, and "H.Uncertainty" to interact with the uncertain elements.
Note that the result H is an uncertain system, called a uss model. The nominal value of H is a state-space (ss) model. Verify that the pole is at -5, as expected from the uncertain parameter's nominal value of 5.
ans = -5
Next, use bodeplot and stepplot to examine the behavior of H. These commands plot the responses of the nominal system and a number of random samples of the uncertain system.
While there are variations in the bandwidth and time constant of H, the high-frequency rolls off at 20 dB/decade regardless of the value of bw. You can capture the more complicated uncertain behavior that typically occurs at high frequencies using the ultidyn uncertain element.
An informal way to describe the difference between the model of a process and the actual process behavior is in terms of bandwidth. It is common to hear "The model is good out to 8 radians/second." The precise meaning is not clear, but it is reasonable to believe that for frequencies lower than, say, 5 rad/s, the model is accurate, and for frequencies beyond, say, 30 rad/s, the model is not necessarily representative of the process behavior. In the frequency range between 5 and 30, the guaranteed accuracy of the model degrades.
The uncertain linear, time-invariant dynamics object ultidyn can be used to model this type of knowledge. An ultidyn object represents an unknown linear system whose only known attribute is a uniform magnitude bound on its frequency response. When coupled with a nominal model and a frequency-shaping filter, ultidyn objects can be used to capture uncertainty associated with the model dynamics.
Suppose that the behavior of the system modeled by H significantly deviates from its first-order behavior beyond 9 rad/s, for example, about 5% potential relative error at low frequency, increasing to 1000% at high frequency where H rolls off. In order to model frequency domain uncertainty as described above using ultidyn objects, follow these steps:
Create a filter W, called the "weight," whose magnitude represents the relative uncertainty at each frequency. The utility makeweight is useful for creating first-order weights with specific low- and high-frequency gains, and specified gain crossover frequency.
Create an ultidyn object Delta with magnitude bound equal to 1.
The uncertain model G is formed by G = Gnom*(1+W*Delta).
If the magnitude of W represents an absolute (rather than relative) uncertainty, use the formula G = Gnom + W*Delta instead.
The following commands carry out these steps:
bw = ureal('bw',5,'Percentage',10); H = tf(1,[1/bw 1]); Gnom = H; W = makeweight(.05,9,10); Delta = ultidyn('Delta',[1 1]); G = Gnom*(1+W*Delta)
G = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 2 states. The model uncertainty consists of the following blocks: Delta: Uncertain 1x1 LTI, peak gain = 1, 1 occurrences bw: Uncertain real, nominal = 5, variability = [-10,10]%, 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.
Note that the result G is also an uncertain system, with dependence on both Delta and bw. You can use bode to make a Bode plot of 20 random samples of G's behavior over the frequency range [0.1 100] rad/s.
In the next section, you design and compare two feedback controllers for G.