We have seen in the previous example, that the estimated parameters converges close to the actual values. A question that often arises is, whether we can ensure that the estimation error vanishes. For the estimator based adaptive controllers briefly presented
3.5. Parameter Convergence
in Subsection 3.2.2, it is common to proof the parameter convergence. However, it is not always possible to proof the stability of the closed loop including estimator and controller.
For the Lyapunov-based approaches, the situation is a little bit different. The conver- gence of the states to the origin is commonly ensured by the design procedure itself. However, in general the parameter convergence cannot be guaranteed as it can be seen from the derivation of the Lyapunov function (3.21).
By using the following state transformation
z(t) = (︄ P12 e(t) Γ−12Θ(t) )︄ (3.31)
with P = P⊤> 0 from (3.17), we can rewrite the error dynamics (3.16) of the adaptive control loop and the estimation error (3.8) to
̇z(t) = J (t) z(t) (3.32)
where the time variant matrix J (t) is given by
J (t) = (︄ P12AmP− 1 2 −Γ 1 2P 1 2Bω⊤(x(t)) Γ12ω⊤(x(t))B⊤P 1 2 0 )︄ . (3.33)
Now, if we can show that z(t) from (3.32) converges asymptotically to zero, we will show simultaneously that the parameter estimation error vanishes. This problem was studied in many contributions, see e.g. [7, 8, 101]. It turned out that in order to proof the asymptotic convergence, the regressor signal ω(x(t)) has to satisfy the persistent excitation condition.
Definition 3.4 (persistent excitation (PE)). Consider the time interval I := [t0, ∞] and
a signal ζ(t) : I → Rn×p. The signal is called persistently exciting on a time interval δ,
iff there exists α, δ > 0, such that the following inequality
t+δ
∫︂
t
ζ(τ )⊤ζ(τ ) dτ ≥ α Ip (3.34)
As it has been shown in [99], the trajectories of (3.32) will converge if the regressor satisfies the persistent excitation condition from Definition 3.4 and ω(x(t)) is bounded. Let us now recall the example from the previous section and the relevance of those findings for this actual problem. The PE condition from Definition 3.4 basically states that the regressor signal ω(x(t)) has to be ”rich” enough to ensure the parameter convergence. But what does this actually mean? If we have a look at the time series in Figure 3.9, we can see that the parameter adjustment happens mainly in the time after the steps of the reference signal. For the time between the steps, the estimation almost does not take place. If we consider for example the damping coefficient d from (3.22), we know from the regressor variable (3.27), that the damping force is proportional to the velocity x2. However, whenever the system is at an equilibrium
point, the damping has no effect on the dynamics. Hence no control input is required to compensate the disturbance since it is zero at that point. Consequently, the estimation error of the damping coefficient is irrelevant at that time and makes no adjustment necessary.
This observation is crucial for nearly all adaptive control systems that are designed using the certainty equivalence principle. Here, the main goal is not to identify the actual plant parameters, instead the controller should find a set of parameters that stabilize the closed loop system. As stated before, parameter convergence would require the PE condition to be satisfied which, in general, cannot be ensured since the regressor signal might depend on an external input like the reference for instance. Nevertheless, there are some studies that deal with the absence of the PE condition, see e.g. [31, 102]
4. Sliding Mode Control
In the following chapter a completely different approach of designing a control system, compared to the previous chapter, is presented. Here the focus shall lie on the use of discontinuous control and its advantages and disadvantages compared to conventional and adaptive approaches. We will start with the basic ideas behind sliding-mode control and present some of the latest results in the field of higher-order sliding-mode control.
The concept was first introduced by Utkin, see e.g. [140]. The main idea is to use a discontinuous control signal in order to compensate for external disturbances and/or model uncertainties. It turned out that it is a very powerful method to design a robust controller that is extremely robust and furthermore even able to stabilize the system in finite time. To this point, sliding-mode control has been used in miscellaneous fields of engineering:
Vehicle dynamics require robust strategies for all kinds of problems. Sliding-mode control can be used for on-line estimation of vehicle [137] and road parameters [136], drive train applications [9, 52] and safety critical control [3, 6, 151].
Electric Machines & Converters profit from the switching signal used in sliding-mode control (SMC). While many applications suffer from undesired effects like chat- tering (see Section 4.3), electrical power systems use methods like pulse-width modulation (PWM) to convert a continuous signal to a switched one in order to increase the efficiency of the electrical circuit. Hence it seems natural to use a switching controller like SMC and apply it directly to various kinds of power converters [129, 130] or electric motors [114] and benefit from the robustness properties. Another advantage is demonstrated in [145] where a sliding-mode approach is efficiently used to reduce the electromagnetic emission of a power converter.
Pneumatic Actuators are highly nonlinear and uncertain systems. In authors of [30, 126] reveal that SMC can handle such complex systems while leaving only a few tuning parameters to the application engineer.
Bio Engineering is a topic where the system dynamics are often partially or even completely unknown. As shown in [51] it turns out that SMC can be used where the relative-degree between input and output is highly uncertain. The authors of [142] use HOSM techniques in order to robustly estimate states of a bioreactor in the presence of measurement noise.