SENSITIVITY ANALYSES MAE 461: DYNAMICS AND CONTROLS 13 SENSITIVITY ANALYSES The previous chapters hinted at the importance of the sensitivity of a system to its parameters. As mentioned in Section 4 of Chapter 3, in the absence of feedback, control forces do not really move a system in a highly predictable manner. The feedback acts as a continuously-acting correction of errors resulting from not knowing precisely the system’s parameters, the external disturbances, and even the control forces being applied. The question arises how one can study the sensitivity of a system to gain confidence that a control system design will stabilize a system as intended. Sensitivity analyses can be performed numerically or analytically. The numerical approach consists of repeatedly simulating the response of a system model while varying different parameters, like the system’s physical parameters, the control parameters, and the disturbances. This approach can be extremely effective. Another approach is to gain more insight into the nature of the errors. This is accomplished by analytical methods. The effect of the changes in the system’s parameters can be described bySENSITIVITY ANALYSES MAE 461: DYNAMICS AND CONTROLS relationships that express the changes in the closed-loop decay rates and closed-loop frequencies in terms of changes in the system’s parameters, that is, the physical parameters and the control gains. This section begins with a description of the most important principles that govern linear stability. These principles provide general conditions that guarantee stability of linear systems. Next, we show how a perturbation analysis can be performed to determine formulas for the changes in the closed-loop decay rates and the frequencies of oscillation in terms of changes in the system’s physical parameters. Then, we show how an approach called the root-locus method can be performed to determine the changes in the closed-loop decay rates and the frequencies of oscillation in terms of changes in control gains. 1. Principles of Linear Stability The equations of a dynamical system, as we know, are in general nonlinear. They can be written in the general form (13 – 1) FxxfxM += ),(&&& where M is a symmetric mass matrix (M = MT), f is a nonlinear function of x and x&, and where F is a forcing function. For the purposes of determining the equilibrium position, we let F = 0. At static equilibrium, (13 – 2) ),(00xf0 =SENSITIVITY ANALYSES MAE 461: DYNAMICS AND CONTROLS To linearize Eq. (13 – 1), we perform a Taylor series approximation of f about x0. Thus, (13 – 3) )()(),(),(000xxfxxxf0xfxxf −⎥⎦⎤⎢⎣⎡∂∂+−⎥⎦⎤⎢⎣⎡∂∂+=&&&TT Substituting Eq. (13 – 3) into (13 – 1) yields the linearized equations (13 – 4) FAxxBxM =++&&& where TT⎥⎦⎤⎢⎣⎡∂∂−=⎥⎦⎤⎢⎣⎡∂∂−=xfBxfA& For purposes that will become apparent momentarily, let’s show that any matrix A can be represented as the sum of a symmetric matrix AS and a skew-symmetric matrix AK. Recall that a matrix AS is symmetric if TSSAA = and that a matrix AK is skew-symmetric if .TKKAA −= To show that A can be written as AS + AK, we simply let (13 – 5) )(21TSAAA += )(21TKAAA −= It is easy to verify that these two matrices are symmetric and skew-symmetric, respectively, and that they sum to A. Returning to the linearized equations of motion, Eq. (13 – 4), let’s now write A and B in terms of their symmetric and skew-symmetric parts. We rewrite Eq. (13 – 4) asSENSITIVITY ANALYSES MAE 461: DYNAMICS AND CONTROLS (13 – 6) FxHKxGCxM=++++ )()(&&& where ⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡∂∂−⎥⎦⎤⎢⎣⎡∂∂=⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡∂∂−⎥⎦⎤⎢⎣⎡∂∂−=⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡∂∂−⎥⎦⎤⎢⎣⎡∂∂=⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡∂∂−⎥⎦⎤⎢⎣⎡∂∂−=TTTTxfxfGxfxfCxfxfHxfxfK&&&&21212121 The matrix C is called the damping matrix, G is called the gyroscopic matrix, K is called the stiffness matrix, and H is called the circulatory matrix. We now ask the following principle question: Under what conditions is the linear system described by Eq. (13 – 6) stable? The answer to this fundamental question depends on the definiteness of the matrices in Eq. (13 – 4). By virtue of the fact that mass is always positive, the mass matrix M is positive definite. The matrix M, by definition, is positive definite if, for any x other than 0 (13 – 7) 0>MxxT The more specific question arises how the definiteness properties of the damping matrix C and the stiffness matrix K affect the system’s stability and also what role the gyroscopic matrix G and the circulatory matrix H play in the system’s stability.SENSITIVITY ANALYSES MAE 461: DYNAMICS AND CONTROLS Toward answering this question, first note that for any skew-symmetric matrix G and for any x (13 – 8) 0=GxxT The proof is quite simple. Let .aT=Gxx Since a is a scalar aaaTTTTTT−=−==== GxxxGxGxx )( The only number a for which a = –a is zero! We are now ready to answer the fundamental question about a linear system’s stability. Pre-multiply Eq. (13 – 4) by Tx&to get (13 – 9) FxxHKxxGCxxMxTTTT&&&&&&&=++++ )()( Let’s examine each of the terms in Eq. (13 – 9), from left to right, and notice the following about each one: (1) where212121)(21212121⎟⎠⎞⎜⎝⎛=+=+=+=xMxxMxxMxxMxxMxxMxxMxxMx&&&&&&&&&&&&&&&&&&&&&&&TTTTTTTTTTdtdxMx&&T21 is the system’s kinetic energy (2) 0≥xCx&&T if 0≥C (called positive semi-definite) (3) 0=xGx&&TSENSITIVITY ANALYSES MAE 461: DYNAMICS AND CONTROLS (4) where212121)(21212121⎟⎠⎞⎜⎝⎛=+=+=+=KxxxKxKxxxKxKxxKxxKxxKxxTTTTTTTTTTdtd&&&&&&&KxxT21 is the system’s potential energy (5) HxxT& can have any sign; it is unpredictable (6) FxT& is power Integrating Eq. (13 – 9) with respect to time yields (13 – 10) ∫∫∫+−−=tTtTtTdtdtdtEtE000)0()( FxHxxxCx&&&& where E(t) is the energy in the system, given by (13 – 11) KxxxMxTTtE2121)( +=&& The conditions under which a system is stable can be surmised from Eq. (13 – 10) and Eq. (13 – 11). First look at the expression for energy E(t) in Eq. (13 – 11). If 0≥K it follows that x approaches a constant value when the energy approaches zero. (since we already know that .)0>M The constant value of x is zero if .0>K A