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MUS420 Introduction to Linear State Space Models

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MUS420 Introduction to Linear State Space Models State Space Models MUS420 Equations of motion for any physical system may be Introduction to Linear State Space Models conveniently formulated in terms of its state x(t): Julius O. Smith III ([email protected]) Center for Computer Research in Music and Acoustics (CCRMA) Input Forces u(t) Department of Music, Stanford University Stanford, California 94305 ft Model State x(t) x˙(t) February 5, 2019 Outline R State Space Models x˙(t)= ft[x(t),u(t)] • where Linear State Space Formulation • x(t) = state of the system at time t Markov Parameters (Impulse Response) • u(t) = vector of external inputs (typically driving forces) Transfer Function • ft = general function mapping the current state x(t) and Difference Equations to State Space Models inputs u(t) to the state time-derivative x˙(t) • Similarity Transformations The function f may be time-varying, in general • • t Modal Representation (Diagonalization) This potentially nonlinear time-varying model is • • Matlab Examples extremely general (but causal) • Even the human brain can be modeled in this form • 1 2 State-Space History Key Property of State Vector The key property of the state vector x(t) in the state 1. Classic phase-space in physics (Gibbs 1901) space formulation is that it completely determines the System state = point in position-momentum space system at time t 2. Digital computer (1950s) 3. Finite State Machines (Mealy and Moore, 1960s) Future states depend only on the current state x(t) • and on any inputs u(t) at time t and beyond 4. Finite Automata All past states and the entire input history are 5. State-Space Models of Linear Systems • “summarized” by the current state x(t) 6. Reference: State x(t) includes all “memory” of the system Linear system theory: The state space • approach L.A. Zadeh and C.A. Desoer Krieger, 1979 3 4 Force-Driven Mass Example Forming Outputs Consider f = ma for the force-driven mass: Any system output is some function of the state, and possibly the input (directly): Since the mass m is constant, we can use momentum • ∆ p(t)= mv(t) in place of velocity (more fundamental, y(t) = ot[x(t),u(t)] since momentum is conserved) y(t) x(t ) and p(t ) (or v(t )) define the state of the mass ot • 0 0 0 m at time t0 In the absence of external forces f(t), all future states Input Forces u(t) • are predictable from the state at time t0: ft x˙(t) Model State x(t) p(t) = p(t0) (conservation of momentum) 1 t x(t) = x(t )+ p(τ) dτ, t t 0 m ≥ 0 Zt0 External forces f(t) drive the state to arbitrary points R • Usually the output is a linear combination of state in state space: variables and possibly the current input: t ∆ ∆ p(t) = p(t0)+ f(τ) dτ, t t0, p(t) = mv(t) y(t) = Cx(t)+ Du(t) ≥ Zt0 where C and D are constant matrices of 1 t x(t) = x(t )+ p(τ) dτ, t t linear-combination coefficients 0 m ≥ 0 Zt0 5 6 Numerical Integration State Definition We need a state variable for the amplitude of each Recall the general state-space model in continuous time: physical degree of freedom x˙(t)= ft[x(t),u(t)] Examples: An approximate discrete-time numerical solution is x(t + T ) = x(t )+ T f [x(t ),u(t )] Ideal Mass: n n n n tn n n • 1 for n =0, 1, 2,... (Forward Euler) Energy = mv2 state variable = v(t) ∆ 2 ⇒ Let gtn[x(tn),u(tn)] = x(tn)+ Tn ftn[x(tn),u(tn)]: Note that in 3D we get three state variables (vx,vy,vz) u(tn) Ideal Spring: • gt 1 Energy = kx2 state variable = x(t) x(tn) x(tn + Tn) 2 ⇒ Inductor: Analogous to mass, so current • Capacitor: Analogous to spring, so charge • T (or voltage = charge/capacitance) z− n Resistors and dashpots need no state variables • This is a simple example of numerical integration for assigned—they are stateless (no “memory”) • solving the ODE ODE can be nonlinear and/or time-varying • The sampling interval T may be fixed or adaptive • n 7 8 State-Space Model of a Force-Driven Mass Force-Driven Mass Reconsidered For the simple example of a mass m driven by external Why not include position x(t) as well as velocity v(t) in force f along the x axis: the state-space model for the force-driven mass? v(t) x˙(t) 0 1 x(t) 0 = + f(t) x =0 v˙(t) 0 0 v(t) 1/m f(t) m We might expect this because we know from before that the complete physical state of a mass consists of its velocity v and position x! There is only one energy-storage element (the mass), • and it stores energy in the form of kinetic energy Therefore, we should choose the state variable to be • velocity v =x ˙ (or momentum p = mv = mx˙) Newton’s f = ma readily gives the state-space • formulation: 1 v˙ = f m or p˙ = f This is a first-order system (no vector needed) • 9 10 Force-Driven Mass Reconsidered and Dismissed State Variable Summary Position x does not affect stored energy State variable = physical amplitude for some • • energy-storing degree of freedom 1 2 Em = mv 2 Mechanical Systems: • Velocity v(t) is the only energy-storing degree of State variable for each • freedom – ideal spring (linear or rotational) Only velocity v(t) is needed as a state variable – point mass (or moment of inertia) • The initial position x(0) can be kept “on the side” to times the number of dimensions in which it can move • enable computation of the complete state in RLC Electric Circuits: position-momentum space: • State variable for each capacitor and inductor t In Discrete-Time: x(t) = x(0) + v(τ) dτ • Z0 State variable for each unit-sample delay In other words, the position can be derived from the Continuous- or Discrete-Time: • velocity history without knowing the force history • Dimensionality of state-space = order of the system Note that the external force f(t) can only drive v˙(t). (LTI systems) • It cannot drive x˙(t) directly: x˙(t) 0 1 x(t) 0 = + f(t) v˙(t) 0 0 v(t) 1/m 11 12 Discrete-Time Linear State Space Zero-State Impulse Response Models (Markov Parameters) For linear, time-invariant systems, a discrete-time state-space model looks like a vector first-order Linear State-Space Model: finite-difference model: y(n) = Cx(n)+ Du(n) x(n +1) = A x(n)+ B u(n) x(n +1) = Ax(n)+ Bu(n) y(n) = C x(n)+ D u(n) The zero-state impulse response of a state-space model is where ∆ easily found by direct calculation: Let x(0) =0 and N x(n) R = state vector at time n u = Ipδ(n)= diag(δ(n),...,δ(n)). Then • ∈ u(n)= p 1 vector of inputs h(0) = Cx(0)B + DI δ(0) = D • × p y(n)= q 1 output vector x(1) = A x(0) + BIpδ(0) = B • × A = N N state transition matrix h(1) = CB • × B = N p input coefficient matrix x(2) = A x(1) + B δ(1) = AB • × h(2) = CAB C = q N output coefficient matrix • × x(3) = A x(1) + B δ(1) = A2B D = q p direct path coefficient matrix • × h(3) = CA2B The state-space representation is especially powerful for . n 1 multi-input, multi-output (MIMO) linear systems h(n) = CA − B , n> 0 • time-varying linear systems • (every matrix can have a time subscript n) 13 14 Zero-State Impulse Response (Markov Linear State-Space Model Parameters) Transfer Function Thus, the “impulse response” of the state-space model Recall the linear state-space model: can be summarized as • y(n) = C x(n)+ D u(n) D, n =0 h (n)= n 1 x(n +1) = A x(n)+ B u(n) ( CA − B, n> 0 and its “impulse response” Initial state x(0) assumed to be 0 • D, n =0 h Input “impulse” is u = I δ(n)= diag(δ(n),...,δ(n)) (n) = n 1 • p ( CA − B, n> 0 Each “impulse-response sample” h(n) is a p q • matrix, in general × The transfer function is the z transform of the • impulse response: The impulse-response terms CAnB for n 0 are • ≥ called Markov parameters ∆ ∞ n ∞ n 1 n H(z) = h(n)z− = D + CA − B z− n=0 n=1 X X 1 ∞ 1 n = D + z− C z− A B "n=0 # X The closed-form sum of a matrix geometric series gives 1 H(z)= D + C (zI A)− B − (a p q matrix of rational polynomials in z) × 15 16 If there are p inputs and q outputs, then H(z) is a System Poles • p q transfer-function matrix (or× “matrix transfer function”) Above, we found the transfer function to be Given transfer-function coefficients, many digital filter 1 • H(z)= D + C (zI A)− B realizations are possible (different computing − structures) The poles of H(z) are the same as those of ∆ 1 H (z) =(zI A)− Example: (p =3, q =2) p − By Cramer’s rule for matrix inversion, the denominator 1 polynomial for (zI A)− is given by the determinant: − 1 ∆ 1 1 z− 1 d(z) = zI A z − 1+ z− − 1 0.5z 1 | − | H − Q determinant (z)= 1 − 1 1 2 where denotes the of the square matrix 2+3z− 1+ z− (1 z− ) | | − Q (also written as det(Q).) 1 0.1z 1 1 z 1 (1 0.1z 1)(1 0.2z 1) − − − − − − − − In linear algebra, the polynomial d(z)= zI A is • called the characteristic polynomial for the| matrix− | A The roots of the characteristic polynomial are called • the eigenvalues of A Thus, the eigenvalues of the state transition matrix • A are the system poles Each mode of vibration gives rise to a pole pair • 17 18 Initial-Condition Response Difference Equation to State Space Form Going back to the time domain, we have the linear A digital filter is often specified by its difference equation discrete-time state-space model (Direct Form I).
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