Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology1 1�cChapter 29 Observers, Mo del-based Controllers 29.1 Intro duction In here we deal with the general case where only a subset of the states, or linear combinations of them, are obtained from measurements and are available to our controller. Such a situation is referred to as the output feedback problem. The output is of the form y � C x + Du : (29.1) We shall examine a class of output feedback controllers constructed in two stages: 1. building an observer | a dynamic system that is driven by the inputs and the outputs of the plant, and produces an estimate of its state variables� 2. using the estimated state instead of the actual state in a state feedback scheme. The resulting controller is termed an observer-based controller or (for reasons that will become clear) a model-based controller. A diagram of the structure of such a controller is given in Figure 29.1. 29.2 Observers An observer comprises a real-time simulation of the system or plant, driven by the same input as the plant, and by a correction term derived from the di�erence between the actual output of the plant and the predicted output derived from the observer. Denoting the state vector of the observer by x^, we have the following state-space description of the observer: � x^ � Ax^ + B u ; L(y ; y^) � (29.2) where L, the observer gain, is some matrix that will be speci�ed later, and ^ y � C x^ + Du is an estimate of the plant output. The term \model-based" for controllers based on an observer refers to the fact that the observer uses a model of the plant as its core.u � � � x^ observer � F L u^ estimate y � C x of x Z}ZZ controller Figure 29.1: Structure of an observer-based, or mo del-based controller, where L denotes the observer gain and F the state feedback gain. De�ne the error vector as x~ � x ; x^. Given this de�nition, the dynamics of the error are determined by the following error model: � x~ � � x ; � x^ � Ax + B u ; Ax^; B u + L(y ; y^) � A(x ; x^) + L(C x ; Cx^) � (A + LC)x~ : (29.3) In general, x~(0) 6 � 0, so we select an L which makes x~(t), the solution to (29.3), approach zero for large t. As we can see, x~(t) ! 0 as t ! 1 for any x~(0) if and only if (A + LC) is stable. Note that if x~(t) ! 0 as t ! 1 then x^(t) ! x(t) as t ! 1. That is, the state estimates eventually converge to their actual values. A key point is that the estimation error does not depend on what the control inputs are. It should be clear that results on the stability of (A + LC) can be obtained by taking the duals of the results on eigenvalue placement for (A + B F ). What we are exploiting here is the fact that the eigenvalues of (A + LC) are the same as those of (A0 + C0L0). Speci�cally we have the following result: Theorem 29.1 There exists a matrix L such that nY det (�I ; [A + LC]) � (� ; �i) (29.4) i�1 for any arbitrary self-conjugate set of complex numbers �1� : : : � �n 2 C if and only if (C� A ) is observ-able. In the case of a single-output system� i.e c is a row vector, one can obtain a formula that is dual to the feedback matrix formula for pole-assignment. Suppose we want to �nd the matrix L such that A + Lc has the characteristic polynomial �d(�) then the following formula will give the desired result L � ;�d(A)O;1 n enwhere On is the observability matrix de�ned as 2 3 C On � 6664 C A . . 7775 : . C An;1 The above formula is the dual of Ackermann's formula which was obtained earlier. Some remarks are in order: 1. If (C� A) is not observable, then the unobservable modes, and only these, are forced to remain as modes of the error model, no matter how Lis chosen. 2. The pair (C� A) is said to be detectable if its unobservable modes are all stable, because in this case, and only in this case, Lcan be selected to change the location of all unstable modes of the error model to stable locations. 3. Despite what the theorem says we can do, there are goo d practical reasons for being cautious in applying the theorem. Trying to make the error dynamics very fast generally requires large L, but this can accentuate the e�ects of any noise in the measurement of y. If y� C x+ �, where � is a noise signal, then the error dynamics will be driven by a term L�, as you can easily verify. Furthermore, unmodeled dynamics are more likely to cause problems if we use excessively large gains. The Kalman �lter, in the special form that applies to the problem we are considering here, is simply an optimal observer. The Kalman �lter formulation models the measurement noise �as a white Gaussian process, and includes a white Gaussian plant noise term that drives the state equation of the plant. It then asks for the minimum error variance estimate of the state vector of the plant. The optimal solution is precisely an observer, with the gain L� chosen in a particular way (usually through the solution of an algebraic Riccati equation). The measurement noise causes us to not try for very fast error dynamics, while the plant noise acts as our incentive for maintaining a goo d estimate (because the plant noise continually drives the state away from where we want it to be). 4. Since we are directly observing plinear combinations of the state vector via y� C x, it might seem that we could attempt to estimate just n; pother (independent) linear combinations of the state vector, in order to reconstruct the full state. One might think that this could be done with an observer of order n; prather than the nthat our full-order observer takes. These expectations are indeed ful�lled in what is known as the Luenberger/Gopinath reduced-order observer. We leave exploration of associated details to some of the homework problems. With noisy measurements, the full-order observer (or Kalman �lter) is to be preferred, as it provides some �ltering action, whereas the reduced-order observer directly presents the un�ltered noise in certain directions of the ^ space.x 29.3 Mo del-Based Controllers Figure 29.2 shows the model-based controller in action, with the observer's state estimate being fed back through the (previously chosen) state feedback gain F. Note that, for this model-based controller, the order of the plant and controller are the same. The numb er of