WIP: External package for matrix equations

README.md

@@ -68,7 +68,7 @@ ss, tf, zpk
 ##### Analysis
 pole, tzero, norm, hinfnorm, linfnorm, ctrb, obsv, gangoffour, margin, markovparam, damp, dampreport, zpkdata, dcgain, covar, gram, sigma, sisomargin
 ##### Synthesis
-care, dare, dlyap, lqr, dlqr, place, leadlink, laglink, leadlinkat, rstd, rstc, dab, balreal, baltrunc
+arec, ared, lyapc, lyapd, lqr, dlqr, place, leadlink, laglink, leadlinkat, rstd, rstc, dab, balreal, baltrunc
 ###### PID design
 pid, stabregionPID, loopshapingPI, pidplots
 ##### Time and Frequency response

src/ControlSystems.jl

@@ -103,11 +103,15 @@ import Base: +, -, *, /, (==), (!=), isapprox, convert, promote_op
 import Base: getproperty, getindex
 import Base: exp # for exp(-s)
 import LinearAlgebra: BlasFloat
-export lyap # Make sure LinearAlgebra.lyap is available
+import ControlMatrixEquations: ared, arec, lyapc, lyapd
+export lyapd, lyapc
+export ared, arec
 import Printf, Colors
 import DSP: conv
 using MacroTools
 
+
+
 abstract type AbstractSystem end
 
 include("types/TimeEvolution.jl")
@@ -169,6 +173,10 @@ include("plotting.jl")
 @deprecate norminf hinfnorm
 @deprecate diagonalize(s::AbstractStateSpace, digits) diagonalize(s::AbstractStateSpace)
 
+@deprecate dlyap lyapd
+@deprecate dare ared
+@deprecate care arec
+
 function covar(D::Union{AbstractMatrix,UniformScaling}, R)
     @warn "This call is deprecated due to ambiguity, use covar(ss(D), R) or covar(ss(D, Ts), R) instead"
     D*R*D'

src/demo_systems.jl

@@ -123,4 +123,21 @@ function woodberry()
 end
 
 
+
+"""
+    `ssfir(c [,Ts=1]) -> sysd`
+
+Returns a (discrete-time) state-space realization `sysd` of a
+finite-impulse response (FIR) system with impulse response `c`.
+"""
+function ssfir(c::AbstractVector{T}, Ts=1) where T
+    n = length(c)-1
+    if n == 0
+        return ss(c[1], Ts)
+    end
+    A = diagm(-1=>ones(T, n-1))
+    B = [1; zeros(T,n-1)]
+    ss(A, B, transpose(c[2:end]), c[1], Ts)
+end
+
 end

src/matrix_comps.jl

@@ -7,75 +7,65 @@ Algorithm taken from:
 Laub, "A Schur Method for Solving Algebraic Riccati Equations."
 http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
 """
-function care(A, B, Q, R)
-    G = try
-        B*inv(R)*B'
-    catch y
-        if y isa SingularException
-            error("R must be non-singular in care.")
-        else
-            throw(y)
-        end
-    end
-
-    Z = [A  -G;
-        -Q  -A']
-
-    S = schur(Z)
-    S = ordschur(S, real(S.values).<0)
-    U = S.Z
-
-    (m, n) = size(U)
-    U11 = U[1:div(m, 2), 1:div(n,2)]
-    U21 = U[div(m,2)+1:m, 1:div(n,2)]
-    return U21/U11
-end
-
-"""`dare(A, B, Q, R)`
-
-Compute `X`, the solution to the discrete-time algebraic Riccati equation,
-defined as A'XA - X - (A'XB)(B'XB + R)^-1(B'XA) + Q = 0, where Q>=0 and R>0
-
-Algorithm taken from:
-Laub, "A Schur Method for Solving Algebraic Riccati Equations."
-http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
-"""
-function dare(A, B, Q, R)
-    if !issemiposdef(Q)
-        error("Q must be positive-semidefinite.");
-    end
-    if (!isposdef(R))
-        error("R must be positive definite.");
-    end
-
-    n = size(A, 1);
-
-    E = [
-        Matrix{Float64}(I, n, n) B*(R\B');
-        zeros(size(A)) A'
-    ];
-    F = [
-        A zeros(size(A));
-        -Q Matrix{Float64}(I, n, n)
-    ];
-
-    QZ = schur(F, E);
-    QZ = ordschur(QZ, abs.(QZ.alpha./QZ.beta) .< 1);
+# function care(A, B, Q, R)
+#     G = try
+#         B*inv(R)*B'
+#     catch y
+#         if y isa SingularException
+#             error("R must be non-singular in care.")
+#         else
+#             throw(y)
+#         end
+#     end
+#
+#     Z = [A  -G;
+#         -Q  -A']
+#
+#     S = schur(Z)
+#     S = ordschur(S, real(S.values).<0)
+#     U = S.Z
+#
+#     (m, n) = size(U)
+#     U11 = U[1:div(m, 2), 1:div(n,2)]
+#     U21 = U[div(m,2)+1:m, 1:div(n,2)]
+#     return U21/U11
+# end
+#
+# """`dare(A, B, Q, R)`
+#
+# Compute `X`, the solution to the discrete-time algebraic Riccati equation,
+# defined as A'XA - X - (A'XB)(B'XB + R)^-1(B'XA) + Q = 0, where Q>=0 and R>0
+#
+# Algorithm taken from:
+# Laub, "A Schur Method for Solving Algebraic Riccati Equations."
+# http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
+# """
+# function dare(A, B, Q, R)
+#     if !ishermitian(Q)# !issemiposdef(Q)
+#         error("Q must be Hermitian");
+#     end
+#     if (!isposdef(R))
+#         error("R must be positive definite");
+#     end
+#
+#     n = size(A, 1);
+#
+#     E = [
+#         Matrix{Float64}(I, n, n) B*(R\B');
+#         zeros(size(A)) A'
+#     ];
+#     F = [
+#         A zeros(size(A));
+#         -Q Matrix{Float64}(I, n, n)
+#     ];
+#
+#     QZ = schur(F, E);
+#     QZ = ordschur(QZ, abs.(QZ.alpha./QZ.beta) .< 1);
+#
+#     return QZ.Z[(n+1):end, 1:n]/QZ.Z[1:n, 1:n];
+# end
 
-    return QZ.Z[(n+1):end, 1:n]/QZ.Z[1:n, 1:n];
-end
-
-"""`dlyap(A, Q)`
 
-Compute the solution `X` to the discrete Lyapunov equation
-`AXA' - X + Q = 0`.
-"""
-function dlyap(A::AbstractMatrix, Q)
-    lhs = kron(A, conj(A))
-    lhs = I - lhs
-    x = lhs\reshape(Q, prod(size(Q)), 1)
-    return reshape(x, size(Q))
-end
 
 """`gram(sys, opt)`
 
@@ -86,7 +76,7 @@ function gram(sys::AbstractStateSpace, opt::Symbol)
     if !isstable(sys)
         error("gram only valid for stable A")
     end
-    func = iscontinuous(sys) ? lyap : dlyap
+    func = iscontinuous(sys) ? lyapc : lyapd
     if opt === :c
         # TODO probably remove type check in julia 0.7.0
         return func(sys.A, sys.B*sys.B')#::Array{numeric_type(sys),2} # lyap is type-unstable
@@ -162,14 +152,14 @@ function covar(sys::AbstractStateSpace, W)
     if !isstable(sys)
         return fill(Inf,(size(C,1),size(C,1)))
     end
-    func = iscontinuous(sys) ? lyap : dlyap
+
     Q = try
-        func(A, B*W*B')
+        iscontinuous(sys) ? lyapc(A, B*W*B') : lyapd(A, B*W*B')
     catch
         error("No solution to the Lyapunov equation was found in covar")
     end
     P = C*Q*C'
-    if !isdiscrete(sys)
+    if iscontinuous(sys)
         #Variance and covariance infinite for direct terms
         direct_noise = D*W*D'
         for i in 1:size(C,1)

src/synthesis.jl

@@ -34,7 +34,7 @@ plot(t,x, lab=["Position" "Velocity"], xlabel="Time [s]")
 ```
 """
 function lqr(A, B, Q, R)
-    S = care(A, B, Q, R)
+    S = care(A, B, Q, R)[1]
     K = R\B'*S
     return K
 end
@@ -99,7 +99,7 @@ plot(t,x, lab=["Position"  "Velocity"], xlabel="Time [s]")
 ```
 """
 function dlqr(A, B, Q, R)
-    S = dare(A, B, Q, R)
+    S = dare(A, B, Q, R)[1]
     K = (B'*S*B + R)\(B'S*A)
     return K
 end

test/runtests.jl

@@ -11,7 +11,7 @@ eye_(n) = Matrix{Int64}(I, n, n)
 my_tests = [
             "test_timeevol",
             "test_statespace",
-            "test_transferfunction",            
+            "test_transferfunction",
             "test_zpk",
             "test_promotion",
             "test_connections",

test/test_demo_systems.jl

@@ -36,4 +36,26 @@ sys = ssrand(2,2,5,proper=false,stable=true, Ts=0.5)
 sys = ssrand(2,2,5,proper=false,stable=false, Ts=0.5)
 @test !isstable(sys)
 
+
+
+c = [2.5]
+sysd = DemoSystems.ssfir(c)
+y = impulse(sysd, 10)[1]
+@test y[1] == c[1]
+@test y[2:11] == zeros(10)
+
+
+c = [0.1, 2.5]
+sysd = DemoSystems.ssfir(c)
+y = impulse(sysd, 10)[1]
+@test y[1:2] == c[1:2]
+@test y[3:11] == zeros(9)
+
+
+c = [1,2,5,3]
+sysd = DemoSystems.ssfir(c)
+y = impulse(sysd, 10)[1]
+@test y[1:4] == c
+@test y[5:11] == zeros(7)
+
 end

test/test_linalg.jl

@@ -4,34 +4,34 @@ b = [0  1]'
 c = [1 -1]
 r = 3
 
-@test care(a, b, c'*c, r) ≈ [0.5895174372762604 1.8215747248860816; 1.8215747248860823 8.818839806923107]
-@test dare(a, b, c'*c, r) ≈ [240.73344504496302 -131.09928700089387; -131.0992870008943 75.93413176750603]
+@test care(a, b, c'*c, r)[1] ≈ [0.5895174372762604 1.8215747248860816; 1.8215747248860823 8.818839806923107]
+@test dare(a, b, c'*c, r)[1] ≈ [240.73344504496302 -131.09928700089387; -131.0992870008943 75.93413176750603]
 
 ## Test dare method for non invertible matrices
-A = [0 1; 0 0];
-B0 = [0; 1];
-Q = Matrix{Float64}(I, 2, 2);
+A = [0 1; 0 0]
+B0 = [0; 1]
+Q = Matrix{Float64}(I, 2, 2)
 R0 = 1
-X = dare(A, B0, Q, R0);
+X = dare(A, B0, Q, R0)[1]
 # Reshape for matrix expression
 B = reshape(B0, 2, 1)
 R = fill(R0, 1, 1)
 @test norm(A'X*A - X - (A'X*B)*((B'X*B + R)\(B'X*A)) + Q) < 1e-14
 ## Test dare for scalars
-A = 1.0;
-B = 1.0;
-Q = 1.0;
-R = 1.0;
-X0 = dare(A, B, Q, R);
+A = 1.0
+B = 1.0
+Q = 1.0
+R = 1.0
+X0 = dare(A, B, Q, R)[1]
 X = X0[1]
 @test norm(A'X*A - X - (A'X*B)*((B'X*B + R)\(B'X*A)) + Q) < 1e-14
 # Test for complex matrices
 I2 = Matrix{Float64}(I, 2, 2)
-@test dare([1.0 im; im 1.0], I2, I2, I2) ≈ [1 + sqrt(2) 0; 0 1 + sqrt(2)]
+@test dare([1.0 im; im 1.0], I2, I2, I2)[1] ≈ [1 + sqrt(2) 0; 0 1 + sqrt(2)]
 # And complex scalars
-@test dare(1.0, 1, 1, 1) ≈ fill((1 + sqrt(5))/2, 1, 1)
-@test dare(1.0im, 1, 1, 1) ≈ fill((1 + sqrt(5))/2, 1, 1)
-@test dare(1.0, 1im, 1, 1) ≈ fill((1 + sqrt(5))/2, 1, 1)
+@test dare(1.0, 1, 1, 1)[1] ≈ fill((1 + sqrt(5))/2, 1, 1)
+@test dare(1.0im, 1, 1, 1)[1] ≈ fill((1 + sqrt(5))/2, 1, 1)
+@test dare(1.0, 1im, 1, 1)[1] ≈ fill((1 + sqrt(5))/2, 1, 1)
 
 ## Test gram, ctrb, obsv
 a_2 = [-5 -3; 2 -9]
@@ -109,6 +109,16 @@ D_static = ss([0.704473 1.56483; -1.6661 -2.16041], 0.07)
 @test norm(D_221) ≈ 3.4490083195926426
 @test norm(ss([1],[2],[3],[4])) == Inf
 
+
+# Test discrete time 2 norms
+c = [1,2,-1,3,5,-2]
+@test norm(DemoSystems.ssfir(c)) == norm(c)
+c = 1:100
+@test norm(DemoSystems.ssfir(c)) == norm(c)
+c = 1:300 # typically enough to break a "naive" Lyapunov solver
+@test norm(DemoSystems.ssfir(c)) == norm(c)
+
+
 #
 ## Test Hinfinity norm computations
 #