EE363 Winter 2008-09Lecture 13Linear quadratic Lyapunov theory• the Lyapunov equation• Lyapunov stability conditions• the Lyapunov operator and integral• evaluating quadratic integrals• analysis of ARE• discrete-time results• linearization theorem13–1The Lyapunov equationthe Lyapunov equation isATP + P A + Q = 0where A, P, Q ∈ Rn×n, and P, Q are symmetricinterpretation: for linear system ˙x = Ax, if V (z) = zTP z, then˙V (z) = (Az)TP z + zTP (Az) = −zTQzi.e., if zTP z is the (generalized)energy, then zTQz is the associated(generalized) dissipationlinear-quadratic Lyapunov theory: linear dynamics, quadratic LyapunovfunctionLinear quadratic Lyapunov theory 13–2we consider system ˙x = Ax, with λ1, . . . , λnthe eigenvalues of Aif P > 0, then• the sublevel sets are ellipsoids (and bounded)• V (z) = zTP z = 0 ⇔ z = 0boundedness condition: if P > 0, Q ≥ 0 then• all trajectories of ˙x = Ax are bounded(this means ℜλi≤ 0, and if ℜλi= 0, then λicorresponds to a Jordanblock of size one)• the ellipsoids {z | zTP z ≤ a} are invariantLinear quadratic Lyapunov theory 13–3Stability conditionif P > 0, Q > 0 then the system ˙x = Ax is (globally asymptotically)stable, i.e., ℜλi< 0to see this, note that˙V (z) = −zTQz ≤ −λmin(Q)zTz ≤ −λmin(Q)λmax(P )zTP z = −αV (z)where α = λmin(Q)/λmax(P ) > 0Linear quadratic Lyapunov theory 13–4An extension based on observability(Lasalle’s theorem for linear dynamics, quadratic function)if P > 0, Q ≥ 0, and (Q, A) observable, then the system ˙x = Ax is(globally asymptotically) stableto see this, we first note that all eigenvalues satisfy ℜλi≤ 0now suppose that v 6= 0, Av = λv, ℜλ = 0then A¯v =¯λ¯v = −λ¯v, soQ1/2v2= v∗Qv = −v∗ATP + P Av = λv∗P v − λv∗P v = 0which implies Q1/2v = 0, so Qv = 0, contradicting observability (by PBH)interpretation: observability condition means no trajectory can stay in the“zero dissipation” set {z | zTQz = 0}Linear quadratic Lyapunov theory 13–5An instability conditionif Q ≥ 0 and P 6≥ 0, then A is not stableto see this, note that˙V ≤ 0, so V (x(t)) ≤ V (x(0))since P 6≥ 0, there is a w with V (w) < 0; trajectory starting at w does notconverge to zeroin this case, the sublevel sets {z | V (z) ≤ 0} (which are unbounded) areinvariantLinear quadratic Lyapunov theory 13–6The Lyapunov operatorthe Lyapunov operator is given byL(P ) = ATP + P Aspecial case of Sylvester operatorL is nonsingular if and only if A and −A share no common eigenvalues,i.e., A does not have pair of eigenvalues which are negatives of each other• if A is stable, Lyapunov operator is nonsingular• if A has imaginary (nonzero, iω-axis) eigenvalue, then Lyapunovoperator is singularthus if A is stable, for any Q there is exactly one solution P of Lyapunovequation ATP + P A + Q = 0Linear quadratic Lyapunov theory 13–7Solving the Lyapunov equationATP + P A + Q = 0we are given A and Q and want to find Pif L yapunov equation is solved as a set of n(n + 1)/2 equations inn(n + 1)/2 variables, cost is O(n6) operationsfast m ethods, that exploit the special structure of the linear equations, cansolve Lyapunov equation with cost O(n3)based on first reducing A to Schur or upper Hessenberg formLinear quadratic Lyapunov theory 13–8The Lyapunov integralif A is stable there is an explicit formula for solution of Lyapunov equation:P =Z∞0etATQetAdtto see this, we note thatATP + P A =Z∞0ATetATQetA+ etATQetAAdt=Z∞0ddtetATQetAdt= etATQetA∞0= −QLinear quadratic Lyapunov theory 13–9Interpretation as cost-to-goif A is stable, and P is (unique) solution of ATP + P A + Q = 0, thenV (z) = zTP z= zTZ∞0etATQetAdtz=Z∞0x(t)TQx(t) dtwhere ˙x = Ax, x(0) = zthus V (z) is cost-to-go from point z (with no input) and integralquadratic cost f unction with matrix QLinear quadratic Lyapunov theory 13–10if A is stable and Q > 0, then for each t, etATQetA> 0, soP =Z∞0etATQetAdt > 0meaning: if A is stable,• we can choo se any positive definite quadratic form zTQz as thedissipation, i.e., −˙V = zTQz• then solve a set of linear equations to find the (unique) quadratic formV (z) = zTP z• V will be positive definite, so it is a Lyapunov f unction that proves A isstablein particular: a linear system is stable if and only if there is a quadraticLyapunov function that proves itLinear quadratic Lyapunov theory 13–11generalization: if A stable, Q ≥ 0, and (Q, A) observable, then P > 0to see this, the L yapunov integral shows P ≥ 0if P z = 0, then0 = zTP z = zTZ∞0etATQetAdtz =Z∞0Q1/2etAz2dtso we conclude Q1/2etAz = 0 for all t ≥ 0this implies that Qz = 0, QAz = 0, . . . , QAn−1z = 0, contradicting(Q, A) observableLinear quadratic Lyapunov theory 13–12Monotonicity of Lyapunov operator inversesuppose ATPi+ PiA + Qi= 0, i = 1, 2if Q1≥ Q2, then f or all t, etATQ1etA≥ etATQ2etAif A is stable, we haveP1=Z∞0etATQ1etAdt ≥Z∞0etATQ2etAdt = P2in other words: if A is stable thenQ1≥ Q2=⇒ L−1(Q1) ≥ L−1(Q2)i.e., inverse Lyapunov operator is monotonic, or preserves matrixinequality, when A is stable(question: is L monotonic?)Linear quadratic Lyapunov theory 13–13Evaluating quadratic integralssuppose ˙x = Ax is stable, and defineJ =Z∞0x(t)TQx(t) dtto fi nd J, we solve Lyapunov equation ATP + P A + Q = 0 for Pthen, J = x(0)TP x(0)in other words: we can evaluate quadratic integral exactly, by solving a setof linear equations, without even computing a matrix exponentialLinear quadratic Lyapunov theory 13–14Controllability and observability Grammiansfor A stable, the controllability Grammian of (A, B) is defined asWc=Z∞0etABBTetATdtand the observability Grammian of (C, A) isWo=Z∞0etATCTCetAdtthese G rammians can be computed by solving the Lyapunov equationsAWc+ WcAT+ BBT= 0, ATWo+ WoA + CTC = 0we always have Wc≥ 0, Wo≥ 0;Wc> 0 if and only if (A, B) is controllable, andWo> 0 if and only if (C, A) is observableLinear quadratic Lyapunov theory 13–15Evaluating a state feedback gainconsider˙x = Ax + Bu, y = Cx, u = Kx, x(0) = x0with closed-loop system ˙x = (A + BK)x stableto evaluate the quadratic integral performance measuresJu=Z∞0u(t)Tu(t) dt, Jy=Z∞0y(t)Ty(t) dtwe solve Lyapunov equations(A + BK)TPu+ Pu(A + BK) + KTK = 0(A + BK)TPy+ Py(A + BK) + CTC = 0then we have Ju= xT0Pux0, Jy= xT0Pyx0Linear quadratic Lyapunov theory 13–16Lyapunov