Control Theory

Most of the functions described in this chapter were contributed by A. Scottedward Hodel @email{A.S.Hodel@eng.auburn.edu} and R. Bruce Tenison @email{Bruce.Tenison@eng.auburn.edu}. They have also written a larger collection of functions for solving linear control problems. It is currently being updated for Octave version 2, with snapshots of the sources available from @url{ftp://ftp.eng.auburn.edu/pub/hodel}.

Function File: [n, m, p] = abcddim (a, b, c, d)

Check for compatibility of the dimensions of the matrices defining the linear system [A, B, C, D] corresponding to

dx/dt = a x + b u
y = c x + d u

or a similar discrete-time system.

If the matrices are compatibly dimensioned, then abcddim returns

n

The number of system states.

m

The number of system inputs.

p

The number of system outputs.

Otherwise abcddim returns n = m = p = -1.

Function File: are (a, b, c, opt)

Return the solution, x, of the algebraic Riccati equation

a' * x + x * a - x * b * x + c = 0

for identically dimensioned square matrices a, b, and c. If b is not square, are attempts to use b*b' instead. If c is not square, are attempts to use c'*c) instead.

To form the solution, Laub's Schur method (IEEE Transactions on Automatic Control, 1979) is applied to the appropriate Hamiltonian matrix.

The optional argument opt is passed to the eigenvalue balancing routine. If it is omitted, a value of "B" is assumed.

Function File: c2d (a, b, t)

Convert the continuous time system described by:

dx/dt = a x + b u

into a discrete time equivalent model

x[k+1] = Ad x[k] + Bd u[k]

via the matrix exponential assuming a zero-order hold on the input and sample time t.

Function File: dare (a, b, c, r, opt)

Return the solution, x of the discrete-time algebraic Riccati equation

a' x a - x + a' x b (r + b' x b)^(-1) b' x a + c = 0

for matrices with dimensions:

a

n by n.

b

n by m.

c

n by n, symmetric positive semidefinite.

r

m by m, symmetric positive definite (invertible).

If c is not square, then the function attempts to use c'*c instead.

To form the solution, Laub's Schur method (IEEE Transactions on Automatic Control, 1979) is applied to the appropriate symplectic matrix.

See also Ran and Rodman, Stable Hermitian Solutions of Discrete Algebraic Riccati Equations, Mathematics of Control, Signals and Systems, Volume 5, Number 2 (1992).

The optional argument opt is passed to the eigenvalue balancing routine. If it is omitted, a value of "B" is assumed.

Function File: dgram (a, b)

Return the discrete controllability or observability gramian for the discrete time system described by

x[k+1] = A x[k] + B u[k]
  y[k] = C x[k] + D u[k]

For example, dgram (a, b) returns the discrete controllability gramian and dgram (a', c') returns the observability gramian.

Function File: [l, m, p, e] = dlqe (a, g, c, sigw, sigv, z)

Construct the linear quadratic estimator (Kalman filter) for the discrete time system

x[k+1] = A x[k] + B u[k] + G w[k]
  y[k] = C x[k] + D u[k] + w[k]

where w, v are zero-mean gaussian noise processes with respective intensities sigw = cov (w, w) and sigv = cov (v, v).

If specified, z is cov (w, v). Otherwise cov (w, v) = 0.

The observer structure is

z[k+1] = A z[k] + B u[k] + k(y[k] - C z[k] - D u[k])

The following values are returned:

l

The observer gain, (a - alc). is stable.

m

The Riccati equation solution.

p

The estimate error covariance after the measurement update.

e

The closed loop poles of (a - alc).

Function File: [k, p, e] = dlqr (a, b, q, r, z)

Construct the linear quadratic regulator for the discrete time system

x[k+1] = A x[k] + B u[k]

to minimize the cost functional

J = Sum (x' Q x + u' R u)

z omitted or

J = Sum (x' Q x + u' R u + 2 x' Z u)

z included.

The following values are returned:

k

The state feedback gain, (a - bk) is stable.

p

The solution of algebraic Riccati equation.

e

The closed loop poles of (a - bk).

Function File: dlyap (a, b)

Solve the discrete-time Lyapunov equation a x a' - x + b = 0 for square matrices a, b. If b is not square, then the function attempts to solve either An Algorithm for Solving the Matrix Equation X = FXF' + S, International Journal of Control, Volume 25, Number 5, pages 745--753 (1977); column-by-column solution method as suggested in Hammerling, Numerical Solution of the Stable, Non-Negative Definite Lyapunov Equation, IMA Journal of Numerical Analysis, Volume 2, pages 303--323 (1982).

Function File: is_controllable (a, b, tol)

Return 1 if the pair (a, b) is controllable. Otherwise, return 0.

The optional argument tol is a roundoff parameter. If it is omitted, a value of 2*eps is used.

Currently, is_controllable just constructs the controllability matrix and checks rank.

Function File: is_observable (a, c, tol)

Return 1 if the pair (a, c) is observable. Otherwise, return 0.

The optional argument tol is a roundoff parameter. If it is omitted, a value of 2*eps is used.

Function File: [k, p, e] = lqe (a, g, c, sigw, sigv, z)

Construct the linear quadratic estimator (Kalman filter) for the continuous time system

dx
-- = a x + b u
dt

y = c x + d u

where w and v are zero-mean gaussian noise processes with respective intensities

sigw = cov (w, w)
sigv = cov (v, v)

The optional argument z is the cross-covariance cov (w, v). If it is omitted, cov (w, v) = 0 is assumed.

Observer structure is dz/dt = A z + B u + k (y - C z - D u)

The following values are returned:

k

The observer gain, (a - kc) is stable.

p

The solution of algebraic Riccati equation.

e

The vector of closed loop poles of (a - kc).

Function File: [k, p, e] = lqr (a, b, q, r, z)

construct the linear quadratic regulator for the continuous time system

dx
-- = A x + B u
dt

to minimize the cost functional

      infinity
      /
  J = |  x' Q x + u' R u
     /
    t=0

z omitted or

      infinity
      /
  J = |  x' Q x + u' R u + 2 x' Z u
     /
    t=0

z included.

The following values are returned:

k

The state feedback gain, (a - bk) is stable.

p

The stabilizing solution of appropriate algebraic Riccati equation.

e

The vector of the closed loop poles of (a - bk).

Function File: lyap (a, b, c)

Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart algorithm (Communications of the ACM, 1972).

If a, b, and c are specified, then lyap returns the solution of the Sylvester equation

a x + x b + c = 0

If only (a, b) are specified, then lyap returns the solution of the Lyapunov equation

a' x + x a + b = 0

If b is not square, then lyap returns the solution of either

a' x + x a + b' b = 0

or

a x + x a' + b b' = 0

whichever is appropriate.

Function File: tzero (a, b, c, d, opt)

Compute the transmission zeros of [A, B, C, D].

The optional argument opt is passed to the eigenvalue balancing routine. If it is omitted, a value of "B" is assumed.

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