Topic #7 16.31 Feedback Control State-Space Systems What are state-space models? Why should we use them? and how do we develop a state-space mode( &ased in classical control design How are they related to the transfer functions What are the basic properties of a state-space model, and how do we analyze these? Copyright [2001 by JOnathan dHow. O
Topic #7 16.31 Feedback Control State-Space Systems • What are state-space models? • Why should we use them? • How are they related to the transfer functions used in classical control design and how do we develop a state-space model? • What are the basic properties of a state-space model, and how do we analyze these? Copyright 2001 by Jonathan How. 1
Fall 2001 16.317-1 troduction State space model: a representation of the dynamics of an Nth order system as a first order differential equation in an N-vector, which is called the state Convert the Mtn order differential equation that governs the dy- namics into N first-order differential equations Classic example: second order mass-spring system mp+ cp+kp= F Let 1=p, then 2=p=1, and (F kn p-kp)/m (F-ca2-ka1)/m Lp k/m-c/m]p][1/ Let u= f and introduce the state →i=Ax+Bu If the measured output of the system is the position, then we have that
Fall 2001 16.31 7—1 Introduction • State space model: a representation of the dynamics of an Nth order system as a first order differential equation in an N-vector, which is called the state. — Convert the Nth order differential equation that governs the dynamics into N first-order differential equations • Classic example: second order mass-spring system mp¨ + cp˙ + kp = F — Let x1 = p, then x2 = p˙ = x˙ 1, and x¨2 = ¨p = (F − cp˙ − kp)/m = (F − cx2 − kx1)/m ⇒ p˙ p¨ = 0 1 −k/m −c/m p p˙ + 0 1/m u — Let u = F and introduce the state x = x1 x2 = p p˙ ⇒ x˙ = Ax + Bu — If the measured output of the system is the position, then we have that y = p = ∙ 1 0 ¸ p p˙ = ∙ 1 0 ¸ x1 x2 = cx
Fall 2001 16.317-2 The most general continuous-time linear dynamical system has form (t)=A(t)x(t)+B(t+)(t) y(t)=C(t)ac(t)+D(tu( where t∈ R denotes time a(tER is the state(vector) u(t Rm is the input or control y(t)∈ RP is the output A(t)ERnxn is the dynamics matrix B(t)∈R n×m is the input matrIx C(t)ERPXn is the output or sensor matrix D(t)E RPxm is the feedthrough matrix Note that the plant dynamics can be time-varying e Also note that this is a mimo system We will typically deal with the time-invariant case Linear Time-Invariant(LTI) state dynamics )= Ax(t)+B y(t)= Ca(t)+ Du(t) so that now A, B, C, D are constant and do not depend on t
Fall 2001 16.31 7—2 • The most general continuous-time linear dynamical system has form x˙(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) where: — t ∈ R denotes time — x(t) ∈ Rn is the state (vector) — u(t) ∈ Rm is the input or control — y(t) ∈ Rp is the output — A(t) ∈ Rn×n is the dynamics matrix — B(t) ∈ Rn×m is the input matrix — C(t) ∈ Rp×n is the output or sensor matrix — D(t) ∈ Rp×m is the feedthrough matrix • Note that the plant dynamics can be time-varying. • Also note that this is a MIMO system. • We will typically deal with the time-invariant case ⇒ Linear Time-Invariant (LTI) state dynamics x˙(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) so that now A, B, C, D are constant and do not depend on t
Fall 2001 16.317-3 Basic definitions Linearity- What is a linear dynamical system? A system g is linear with respect to its inputs and output y if superposition holds: G(a11+a22)=a1Gu1+a2G So if y1 Is th he response o of g to u1(91 Gul), and y2 is the response of G to u2(92=Gu2), then the response to a11+a202 IS a191+ a292 A system is said to be time-invariant if the relationship between the input and output is independent of time. So if the response to (t) is yt) then th response to u(t-to is y(t- to (t)is called the state of the system at t because Future output depends only on current state and future input Future output depends on past input only through current state State summarizes effect of past inputs on future output-like the memory of the system Example: Rechargeable fashlight- the state is the current state oj charge of the battery. If you know that state, then you do not need to know how that level of charge was achieved (assuming a perfect battery)to predict the future performance of the fashlight
Fall 2001 16.31 7—3 Basic Definitions • Linearity — What is a linear dynamical system? A system G is linear with respect to its inputs and output u(t) → G(s) → y(t) if superposition holds: G(α1u1 + α2u2) = α1Gu1 + α2Gu2 So if y1 is the response of G to u1 (y1 = Gu1), and y2 is the response of G to u2 (y2 = Gu2), then the response to α1u1 + α2u2 is α1y1 + α2y2 • A system is said to be time-invariant if the relationship between the input and output is independent of time. So if the response to u(t) is y(t), then the response to u(t − t0) is y(t − t0) • x(t) is called the state of the system at t because: — Future output depends only on current state and future input — Future output depends on past input only through current state — State summarizes effect of past inputs on future output — like the memory of the system • Example: Rechargeable flashlight — the state is the current state of charge of the battery. If you know that state, then you do not need to know how that level of charge was achieved (assuming a perfect battery) to predict the future performance of the flashlight
Fall 2001 16.317-4 Creating Linear State-Space Models o most easily created from Ntn order differential equations that de- scribe the dynamics This was the case done before Only issue is which set of states to use -there are many choices Can be developed from transfer function model as well Much more on this later e Problem is that we have restricted ourselves here to linear state space models, and almost all systems are nonlinear in real-life Can develop linear models from nonlinear system dynamics
Fall 2001 16.31 7—4 Creating Linear State-Space Models • Most easily created from Nth order differential equations that describe the dynamics — This was the case done before. — Only issue is which set of states to use — there are many choices. • Can be developed from transfer function model as well. — Much more on this later • Problem is that we have restricted ourselves here to linear state space models, and almost all systems are nonlinear in real-life. — Can develop linear models from nonlinear system dynamics