17 MODAL ANALYSIS 17.1 Introduction The evolution of states in a linear system occurs through independent modes, which can be driven by external inputs, and observed through plant output. This section provides the basis for modal analysis of systems. Throughout, we use the state-space description of a system with D = 0: γx˙= Aγx + Bγu y x.γ = Cγ17.2 Matrix Exponential 17.2.1 Definition In the instance of an unforced response to initial conditions, consider the system γ˙x = Aγx, γx(t = 0) = γν. In the scalar case, the response is x(t) = νeat , giving a decaying exponential if a < 0. The same notation holds for the case of a vector γx, and matrix A: At γγx(t) = e ν, where (At)2 At e = I + At + + e2! · · · At is usually called the matrix exponential.� � 84 17 MODAL ANALYSIS 17.2.2 Modal Canonical Form Introductory material on the eigenvalue problem and modal decomposition can be found in the MATH FACTS section. This modal decomposition of A leads to a very useful state-space representation. Namely, since A = V ΦV −1, a transformation of state variables can be made, γx = V γz, leading to γ˙z = Φγz + V −1Bγu (197) y z. γ = CV γThis is called the modal canonical form, since the states are simply the modal amplitudes. These states are uncoupled in Φ, but may be coupled through the input (V −1B) and output (CV ) mappings. The modal form is numerically robust for computations. 17.2.3 Modal Decomposition of Response Now we are ready to look at the matrix exponential eAt in terms of its constituent modes. Employing the above form for A, we find that (At)2 At e = I + At + + 2! ··· (Φt)2 WTV I + Φt + += ··· 2! = V e �tWT n ⎫ = e vi γT�i tγ wi . i=1 In terms of the response to an initial condition γν, we have n ⎫�itγvi( γT wγx(t) = γν).e i i=1 The product γT ν is a scalar, the projection of the initial conditions onto the i’th mode. wi γIf γ wi, then the product is zero and the i’th mode does not respond. ν is perpendicular to γOtherwise, the i’th mode does participate in the response. The projection of the i’th mode onto the states γx is through the right eigenvector γvi. For stability of the system, the eigenvalues of A, that is, ηi, must have negative real parts; they are in fact the poles of the equivalent transfer function description. 17.3 Forced Response and Controllability Now consider the system with an external input γu:17.4 Plant Output and Observability 85 γx˙= Aγx + Bγu, γx(t = 0) = γν. Taking the Laplace transform of the system, taking into account the initial condition for the derivative, we have sγx(s) − γν = Aγx(s) + Bγu(s) −∀ γx(s) = (sI − A)−1 γν + (sI − A)−1Bγu(s). AtThus (sI − A)−1 can be recognized as the Laplace transform of the matrix exponential e . In the time domain, the second term then has the form of a convolution of the matrix exponential and the net input Bγu: � t γx(t) = e A(t−φ )Bγu(φ )dφ 0 n ⎫� t = e vi γT�i (t−φ )γ wi Bγu(φ )dφ. 0i=1 Suppose now that there are m inputs, such that B = [γb1,γ γbm]. Then some rearrange-b2, ···, ment will give n ⎫ m ⎫� t γx(t) = γvi ( γw Tγbk ) e �i (t−φ )γuk(φ )dφ. i 0i=1 k=1 The product γTγbk , a scalar, represents the projection of the k’th control channel onto the wi i’th mode. We say that the i’th mode is controllable from the k’th input if the product is nonzero. If a given mode i has γTγ= 0 for all input channels k, then the mode is wi bk uncontrollable. In normal applications, controllability for the entire system is checked using the following test: Construct the so-called controllability matrix: Mc = [B, AB, ···, An−1B]. (198) This matrix has size n ×(nm), where m is the number of input channels. If Mc has rank n, then the system is controllable, i.e., all modes are controllable. 17.4 Plant Output and Observability We now turn to a related question: can the complete state vector of the system be observed y, and the known control γgiven only the output measurements γ u? The response due to the external input is easy to compute deterministically, through the convolution integral. Consider the part due to initial conditions γν. We found above� � 86 18 CONTROL SYSTEMS – LOOPSHAPING n ⎫ vi γT�i tγ wγx(t) = γν.e i i=1 The observation is γ x (r channels of output), and writing y = Cγ⎭γc1 T ⎛⎝ ⎞⎠C = .· Tγcr the k’th channel of the output is n ⎫T �i t T yk(t) = (γck γvi)e ( γw γν).i i=1 The i’th mode is observable in the k’th output if the product γckT γvi ≥= 0. We say that a system is observable if every mode can be seen in at least one output channel. The usual test for system observability requires computation of the observability matrix: CT , AT CT )n−1CT � , ···, (AT . Mo = (199) This matrix has size n × (rn); the system is observable if Mo has rank