Continuous Dynamics Solving LTI state-space equations גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון

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1 Continuous Dynamics Solving LTI state-space equations גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון

2 2 State Space Models For a causal system with m inputs u t R m and p outputs y t R p, an nth-order state-space description is obtained by introducing n latent variables x t R n called state variables In order to obtain a description of the form: 1. An equation for updating x (usually also takes u into account) 2. An equation for mapping values of x to values of y (sometimes also influenced by u)

3 3 Linear Time Invariant State Space Models For continuous time systems: dx/dt = fax x t t +, ubu(t) t, t y(t) = Cx g xt t +, udu(t) t, t For discrete time systems: x[k x[t + 1] 1] = fax[t] d x k +, ubu[t] k, k y[k] y[t] = gcx[t] d x[k], + u[k], Du[t] k

4 4 Example: One-Dimensional Movement of an Object The Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring: where mξ (t) = u(t) k 1 ξ (t) k 2 ξ(t) ξ(t) is position; ξ (t) is velocity; ξ (t) is acceleration F(t) is an applied force k 1 is the viscous friction coefficient k 2 is the spring constant m is the mass of the object

5 5 Example: One-Dimensional Movement of an Object m ξ (t) = u(t) k 1 ξ (t) k 2 ξ(t) x 1 t x 2 t = 0 1 k 2 /m k 1 /m x 1 t x 2 t + 0 1/m u(t) y(t) = 1 0 x 1 t x 2 t u x = Ax + Bu y = Cx + Du y x 1 t = ξ(t) represents the position of the object x 2 (t) = x 1(t) is the velocity of the object x 2(t) = x 1 t is the acceleration of the object The input u t is the force applied to the object The output y(t) is the position of the object

6 6 Practice Exercise Prove that the explicit solution to discrete-time linear state equation is: x[t + 1] = Ax[t] + Bu[t] y[t] = Cx[t] + Du[t] t 1 x t = A t x t 0 + A t 1 τ Bu[τ] τ=0

7 7 The Matrix Exponential A key quantity in determining solutions to continuous-time LTI state equations is the matrix exponential defined as e At = i=0 1 i! Ai t i Practice exercise: 1. Prove that deat dt = Ae At = e At A 2. Does this mean that x t = e At x 0 is a solution to the differential equation x t = Ax t, x 0 = x 0?

8 8 Example: A is nilpotent Let A = It is easy to check that A 2 = 0. Thus, e At = i=0 1 i! Ai t i = I + At = 6t + 1 9t 4t 1 6t

9 9 Example: A is diagnosable Let A = The eigenvectors of A are 1 2 and 1 1. The eigenvalues of A are 2 and 1. Therefore A = T 1 DT where T = and D = Thus, e At = T 1 e Dt T = e 2t ( e t ) e 2t ( +e t ) 2e 2t ( 1 + e t ) e 2t 2 + e t

10 10 Practice Exercise Apply Leibnitz s rule: to prove that is a solution to the linear state equation x t = Ax t + Bu t

11 11 Output Solution Notice that the state and the output solutions consist of two terms each: An initial condition response due to x 0 And a forced response which depends on the input u(t) over the interval [t 0, t] Practice Exercise: Prove that the system is LTI if x 0 = 0

12 12 Linearization Although almost every real system includes nonlinear features, many systems can be reasonably described, at least within certain operating ranges, by linear models

13 13 Linearization

14 14 Linearization in Two Dimensions

15 15 Linearization in Two Dimensions The function at the right hand side of x = f x represents a surface So the local linear representation should geometrically be a plane This is obtained by the Jacobian matrix of the functional form f(x) at an equilibrium point: If the state space is two dimensional, given by equations of the form x = f 1 x, y y = f 2 (x, y) then the local linearization at an equilibrium point δx δy = f 1/ x f 1 / y f 2 / x f 2 / y where δx = x x and δy = y y The matrix containing the partial derivatives is called the Jacobian matrix The partial derivatives are calculated at the equilibrium point δx δy

16 16 Linearization with MATLAB Modern computational packages include special commands to compute linearized models around a user defined (pre-computed) operating point. In the case of MATLAB-Simulink,the appropriate commands are: linmod (for continuous time systems) and dlinmod (for discrete time and hybrid systems).

17 17 Caution! It is obvious that linearized models are only approximate models Thus these models should be used with appropriate caution (as indeed should all models) In the case of linearized models, the next term in the Taylor s series expansion can often be usefully employed to tell us something about the size of the associated modeling error

18 18 Summary of Linearization Linear models often give deep insights and lead to simple control strategies Linear models can be obtained by linearizing a nonlinear model at an operating point Caution is needed to deal with unavoidable modeling errors