Abstract
The parameters of a process may be unknown or may change slowly over time. This chapter discusses how one can control a process with unknown parameters. Adaptive methods adjust parameters in response to information about external inputs and system outputs. Adaptive error-correcting techniques often provide a good approach to coping with unknown nonlinear system dynamics.
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The parameters of a process may be unknown or may change slowly over time. How can one control a process with unknown parameters?
Earlier chapters discussed robust methods. Those methods provide good response to a broad set of alternative process dynamics.
This chapter presents adaptive control, in which the control system adjusts itself by using measurements of the system’s response. I follow the example from Sect. 6.2.4 of Ioannou and Sun (2012).
In adaptive control, the system repeatedly updates the controller parameters to reduce the error between the system’s actual output and the output of an ideal target response model. Figure 11.1 shows the structure of a common approach known as model reference adaptive control.
Suppose the process dynamics are given by the affine form in Eq. 10.3 as
which describes linear systems and also a wide variety of nonlinear systems. In this example, we know the functions f and g, but do not know the parameter values for a and b. The goal is to design a control input, u, that causes the system output, y, to match the output of a specified model.
1 General Model
Typically, one chooses a simple linear model for the design target. In this example, we use
Here, the parameters \(a_m\) and \(b_m\) are known aspects of the target model specification, and r is the reference or external environmental input. For a constant reference input, this model converges to the reference exponentially at rate \(a_m\), with amplitude of the response relative to the input of \(b_m/a_m\). Figure 11.2 illustrates the design target response for a sinusoidal input, r.
For given values of a and b, the control input
transforms the process model in Eq. 11.1 into the target model in Eq. 11.2.
If the parameters a and b are unknown, then the input, u, must be based on the estimates for \(k_1(t)\), \(k_2(t)\), and w(t). The estimates are updated by an adaptive process in response to the error difference between system and model output, \(e=y-y_m\). The dynamics of the error are \(\dot{e}=\dot{y}-\dot{y}_m\).
To obtain an expression for \(\dot{e}\), we need a modified form of \(\dot{y}\) that contains only the known parameters \(a_m\) and \(b_m\) and the estimates \(k_1\), \(k_2\), and w. The first step expresses the process dynamics in Eq. 11.1 by adding and subtracting \(b{\big [}k_1^*f(y)+k_2^*y+w^*r{\big ]}\) and using the identities \(bk_1^*=-a\) and \(bk_2^*=-a_m\) and \(bw^*=b_m\), yielding
Write the tracking errors as \(\tilde{k}_1=k_1-k_1^*\) and \(\tilde{k}_2=k_2-k_2^*\) and \(\tilde{w}=w-w^*\). The error dynamics can then be written as
To analyze the error dynamics, we need expressions for the processes used to update the parameter estimates. A common choice is
in which I have assumed that \(b>0\).
2 Example of Nonlinear Process Dynamics
The general results of the prior section can be applied to any linear process or to any nonlinear process that can be approximated by the affine form of Eq. 11.1. For this nonlinear example, let
with \(f(y)=y^2\) and \(g(y)=1\).
Figure 11.3 illustrates the rate of adaptation for various parameters. As the adaptation parameters, \(\gamma \), increase, the system output converges increasingly rapidly to the target model output.
3 Unknown Process Dynamics
The previous section assumed a particular form for the process dynamics in Eq. 11.4, with unknown parameters a and b. How could we handle a process with unknown dynamics?
One simple approach is to assume a very general form for the process dynamics, such as a polynomial
and then run the adaptation process on the parameters \((a_0,a_1,\ldots ,a_n, b)\). One could use other generic forms for the dynamics and estimate the parameters accordingly. This approach provides a way for the system output to mimic the model output, without the system necessarily converging to use the same mathematical description of dynamics as in the model.
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Frank, S.A. (2018). Adaptive Control. In: Control Theory Tutorial. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-91707-8_11
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DOI: https://doi.org/10.1007/978-3-319-91707-8_11
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