Our system is linear, so let’s write it out in the following state space representation Now that we’ve done that, let’s figure out what the derivatives of x_3 and x_4 are Next let’s define x_3 and x_4 as the derivatives of x_1 and x_2 respectively Let’s use x_i, where i is a number from 1 to 4, and let’s denote the vector of them as X.įirst let’s define x_1 and x_2 as the following We’ll need a change of variables to differentiate the 2 2nd order equations, from the 4 1st order equations. We can always convert m number of nth order differential equations to (m*n) first order differential equations, so let’s do that now. Then, appealing to newton’s second law, we can turn these into two second order equations of motion We can use hooks law to determine the forces acting on the two blocks (don’t forget the force of the second block acting on the first) Where F_s is the force from the spring, K_s is the spring constant, and d is how far away from normal the spring has been stretched. We start every problem with a Free Body Diagram Today, we’ll explore another system that produces Lissajous curves, a double spring-mass system, analyze it, and then simulate it using ODE45.įBD, Equations of Motion & State-Space Representation In layman terms, Lissajous curves appear when an object’s motion’s have two independent frequencies. These are called Lissajous curves, and describe complex harmonic motion. In this form, the characteristic polynomial of the system appearsĮxplicitly in the last column of the A matrix.From orbits around Lagrange Points, to double pendulums, we often run into a family of loopy, beautiful, curves. The observable canonical form of a system is the dual (transpose) of its controllableĬanonical form. Then, create the system with the ss command. H( s), then you can use the coefficients ɑ 0,…, ɑ n–1, β 0,…, β n–1, and d 0 to construct theĬontrollable canonical-form matrices in MATLAB. If you can obtain the system in the transfer-function form There is no MATLAB ® command for directly computing controllable canonical form. Like companion formĪnd observable canonical form, it can be ill-conditioned for computation. This form, the coefficients of the characteristic polynomial appear in the last row ofĪ minimal realization in which all model states are controllable. This form is also known as phase-variable canonical form. Is the dual (transpose) of controllable companion form, as follows:Ī c o n t =, B c o n t =, C c o n t =, D c o n t = d 0. Obsv(H.A,H.B) instead of T = ctrb(H.A,H.B). Observable Companion FormĪ related form is obtained using the observability state transformation T = Hence,Īvoid using it for computation when possible. Matrix, which is almost always numerically singular for mid-range orders. The transformation to companion form is based on the controllability The companion transformation requires that the system be controllable from theįirst input. When performing system identification using commands such as ssest (System Identification Toolbox) or n4sid (System Identification Toolbox), obtain companion form by T = ctrb(H.A,H.B) to put the A matrix into The command canon(H,"companion") computes a controllableĬompanion-form realization of H by using the state transformation This form does not impose a particular structure on the rest ofĭ ccom. įor multi-input systems, A ccom has the sameįorm, and the first column of B ccom is as
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