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23High Gain Observer Introduction Consider the following system where (A,C) is observable.This form is special because g depends only on y and u.Taking the observer aswe obtain that satisfiesTherefore, designing C such that AHC is Hurwitz guarantees asymptoticerror convergence.
24High Gain Observer (Continued) However, any error in modeling g will be reflected in the estimation errorequation. Thus,where is a nominal model of g. HenceWe give a special design of the observer gain that makes the observer robustto uncertainties in modeling the nonlinear functions. The technique, called ashigh-gain observers, works for a wide class of nonlinear systems and guaranteesthat the output feedback controller recovers the performance of the state feedbackcontroller when the observer gain is sufficiently high.
25High Gain Observer (Continued) The main result is a separation principle that allows us to separate the design intotwo tasks. First, design a state feedback controller that stabilizes the system andmeets other design specifications. Then, obtain an output feedback controller byreplacing x by provided by the high-gain observer. A key property that makesthis separation possible is the design of the state feedback controller to be globallybounded in x.
26ExampleEx:Assume that u = (x) is a local state feedback control law that stabilizes the origin.To implement this control law using only y, we use the observerwhere is a nominal model of the nonlinear functionThenwhere
27Example (Continued) We want to design such that In the absence of , asymptotic error convergence is achieved by designingH such thatis Hurwitz. In the presence of , we need to design H with the goal of rejectingthe effect of onThis is ideally achieved, for any , if the transfer functionfrom to is ideally zero.
28Example (Continued)While this is impossible, we can make arbitrarily small bychoosingTakingit can be shown thatHenceDefine the scaled estimation errorsThen the newly defined variables satisfy the singularly perturbed equation.This equation shows clearly that reducing diminishes the effect of .
29Example (Continued) Notice, however, that will be whenever Consequently, the solution contains a term ofIn fact,This behavior is known as the peaking phenomenon.
30Globally Stabilized by State Feedback Controller Let’s consider the systemwhich can be globally stabilized by the state feedback controllerThe output controller is taken aswhere the observer gain assigns the eigen values of
31State Feedback Controller The above figure shows a counter intuitive behavior as 0. Since decreasing causes the estimation error to decay faster toward zero, one would expect theresponse under output feedback to approach the response under state feedback as decreases. This is the impact of peaking phenomenon. Fortunately, we canovercome the peaking phenomenon by saturating the control outside a compactregion of interest in order to create a buffer that protects the plant from peaking.
32State Feedback Controller Suppose the control is saturated asThe above figure shows the performance of the system under saturated stateand output feedback. The control u is shown on a shorter time interval thatexhibits control saturation during peaking. The peaking period decreases with .
33StabilizationConsider the MIMO nonlinear system(1)
35Output Feedback Controller The functions , and q are locally Lipschitz and (0,0,0)=0, (0,0,0)=0,q(0,0)=0. Our goal is to find an output feedback controller to stabilize theorigin.We use a two-step approach to design the output feedback controller.(i) A partial state feedback controller using x and is designedto asymptotically stabilize the origin.(ii) A high-gain observer is used to estimate x from y.The state feedback controller can be shown aswhere r, are locally Lipschitz in their arguments over the domain of interestand globally bounded functions of x. Moreover, r(0,0,0)=0 and (0,0,0)=0.
36Output Feedback Controller (Continued) For convenience, we write the closed-loop system under state feedback as(2)where X = (x, z, ).The output controller is taken as(3)
37Output Feedback Controller (Continued) The observer gain H is chosen as
38TheoremTheorem: Consider the closed-loop system of the plant (1) and the outputfeedback controller (3). Suppose the origin of (2) is asymptoticallystable and R is its region of attraction. Let S be any compact set in theinterior of R and Q be any compact subset of Then,
40ResultsThe theorem shows that the output feedback controller recovers theperformance of the state feedback controller for sufficiently small .Note that(i) recovery of exponential stability(ii) recovery of the region of attraction in the sensethat we can recover any compact set in its interior(iii) the solution under output feedback approaches thesolution under the state feedback as 0.
41Example Ex: Consider the following plant: Note : the given system is in triangular form. Thus, it is stabilizablebyThe output feedback controller is
42Example (Continued) with For simulation: All system eigenvalues at 5. All observer eigenvalues at 5/ with = 0.05.