Lecture – 33 Stability Analysis of Nonlinear Systems Using Lyapunov Theory – I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline z Motivation z Definitions z Lyapunov Stability Theorems z Analysis of LTI System Stability z Instability Theorem z Examples ADVANCED CONTROL SYSTEM DESIGN 2 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore References z H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003. z J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991. z H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996. ADVANCED CONTROL SYSTEM DESIGN 3 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Techniques of Nonlinear Control Systems Analysis and Design z Phase plane analysis z Differential geometry (Feedback linearization) z Lyapunov theory z Intelligent techniques: Neural networks, Fuzzy logic, Genetic algorithm etc. z Describing functions z Optimization theory (variational optimization, dynamic programming etc.) ADVANCED CONTROL SYSTEM DESIGN 4 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Motivation z Eigenvalue analysis concept does not hold good for nonlinear systems. z Nonlinear systems can have multiple equilibrium points and limit cycles. z Stability behaviour of nonlinear systems need not be always global (unlike linear systems). z Need of a systematic approach that can be exploited for control design as well. ADVANCED CONTROL SYSTEM DESIGN 5 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions System Dynamics XfX =→() fD:Rn (a locally Lipschitz map) D : an open and connected subset of Rn Equilibrium Point ( X e ) XfXee= ( ) = 0 ADVANCED CONTROL SYSTEM DESIGN 6 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions Open Set A set A ⊂ n is open if for every pA∈ , ∃⊂ Bpr () A Connected Set z A connected set is a set which cannot be represented as the union of two or more disjoint nonempty open subsets. z Intuitively, a set with only one piece. Space A is connected, B is not. ADVANCED CONTROL SYSTEM DESIGN 7 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions Stable Equilibrium X e is stable, provided for each ε >∃0,δε ( ) > 0 : X (0)−<XXtXtteeδε ( ) ⇒ ( ) −<∀≥ ε 0 Unstable Equilibrium If the above condition is not satisfied, then the equilibrium point is said to be unstable ADVANCED CONTROL SYSTEM DESIGN 8 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions Convergent Equilibrium If ∃−<⇒=δδ:XX( 0) ee lim XtX( ) t→∞ Asymptotically Stable If an equilibrium point is both stable and convergent, then it is said to be asymptotically stable. ADVANCED CONTROL SYSTEM DESIGN 9 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions Exponentially Stable −λt ∃>αλ,0:Xt( ) −≤ Xee α X( 0) − X e ∀> t 0 wheneverXX() 0 −<e δ Convention The equilibrium point X e = 0 (without loss of generality) ADVANCED CONTROL SYSTEM DESIGN 10 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions A function VD: → R is said to be positive semi definite in D if it satisfies the following conditions: ()0i∈ D and V (0)= 0 ()ii V() X≥∀∈ 0, X D VD: → R is said to be positive definite in D if condition (ii) is replaced by VX( ) >−0{0} inD VD: → R is said to be negative definite (semi definite) in D if −VX()is positive definite. ADVANCED CONTROL SYSTEM DESIGN 11 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Lyapunov Stability Theorems Theorem – 1 (Stability) Let XXfXfD==→0 be an equilibrium point of ( ) , :Rn . Let VD:→ R be a continuously differentiable function such that: ()iV() 0= 0 ()ii V() X>− 0, in D {0} ()iii V () X≤− 0, in D {0} Then X = 0 is "stable". ADVANCED CONTROL SYSTEM DESIGN 12 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Lyapunov Stability Theorems Theorem – 2 (Asymptotically stable) Let XXfXfD==→0 be an equilibrium point of ( ) , :Rn . Let VD:→ R be a continuously differentiable function such that: ()iV() 0= 0 ()ii V() X>− 0, in D {0} ()iii V () X<− 0, in D {0} Then X = 0 is "asymptotically stable". ADVANCED CONTROL SYSTEM DESIGN 13 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Lyapunov Stability Theorems Theorem – 3 (Globally asymptotically stable) Let XXfXfD==→0 be an equilibrium point of ( ) , :Rn . Let VD:→ R be a continuously differentiable function such that: ()iV() 0= 0 ()ii V() X>− 0, in D {0} (iii ) V() X is "radially unbounded" ()iv V () X<− 0, in D {0} Then X = 0 is "globally asymptotically stable". ADVANCED CONTROL SYSTEM DESIGN 14 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Lyapunov Stability Theorems Theorem – 3 (Exponentially stable) Suppose all conditions for asymptotic stability are satisfied. In addition to it, suppose ∃ constants kkk123, , , p : pp ()ikX12≤≤ VXkX() p ()ii V() X≤− k3 X Then the origin X = 0 is "exponentially stable". Moreover, if these conditions hold globally, then the origin X = 0 is "globally exponentially stable". ADVANCED CONTROL SYSTEM DESIGN 15 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Example: Pendulum Without Friction z System dynamics ⎡⎤x1 ⎡ x 2 ⎤ = ⎢ ⎥ ⎢⎥x − gl/sin x xx12θ , θ ⎣⎦2 ⎣ () 1 ⎦ z Lyapunov function VKEPE= + 1 2 =+ml()ω mgh 2 1 =+−ml22 x mg(1 cos x ) 2 21 ADVANCED CONTROL SYSTEM DESIGN 16 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Pendulum Without Friction T VX ()(=∇ V ) fX () ⎡⎤∂∂VV T = ⎢⎥⎣⎦⎡⎤fX12() fX () ⎣⎦∂∂xx12 T ⎡⎤g ⎡⎤2 =−⎣⎦mglsin x1221 ml x x sin x ⎣⎦⎢⎥l =−=mglx21sin x mglx 21 sin x 0 VX ()≤ 0(nsdf) Hence, it is a “stable” system. ADVANCED CONTROL SYSTEM DESIGN 17 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Pendulum With Friction Modify the previous example by adding the friction force klθ ma = −mg sinθ − klθ Defining the same state variables as above xx12= gk x =−sin xx − 212lm ADVANCED CONTROL SYSTEM DESIGN 18 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Pendulum With Friction T VX ()(=∇ V ) fX () ⎡⎤∂∂VV T = ⎢⎥⎣⎦⎡⎤fX12() fX () ⎣⎦∂∂xx12 T ⎡ gk⎤ ⎡⎤2 =−−⎣⎦mglsin x122 ml x x sin x 12 x ⎣⎢ lm⎦⎥ 22 =−kl x2 VX ()≤ 0nsdf ( ) Hence, it is also just a “stable” system. (A frustrating result..!) ADVANCED CONTROL SYSTEM DESIGN 19 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Analysis of Linear Time Invariant System System dynamics: XAXA =∈, Rnn× Lyapunov function: VX( ) => XPXT ,0 P (pdf) Derivative analysis: VXPXXPX=+TT =+X TTAPX X T PAX TT =+X ()AP PAX ADVANCED CONTROL SYSTEM DESIGN 20 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Analysis of Linear Time Invariant System For stability, we aim for VXQXQ =−T ( >0) TT T By comparing X ( AP+=− PAX) XQX For a non-trivial solution PA+ AT P+= Q 0 (Lyapunov Equation) ADVANCED CONTROL SYSTEM DESIGN 21 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Analysis of Linear Time Invariant System nn× Theorem : The eigenvalues λλii of a matrix A∈< satisfy Re( ) 0 if and only if for any given symmetric pdf matrix Q, ∃ a unique pdf matrix P satisfying the Lyapunov equation. Proof: Please see Marquez book, pp.98-99. Note : PQ and are related to each other by the following relationship: ∞ T PeQedt= ∫ At At 0 However, the above equation is seldom used to compute PP. Instead is directly solved from the Lyapunov equation. ADVANCED CONTROL SYSTEM DESIGN 22 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Analysis of Linear Time Invariant Systems z Choose an arbitrary symmetric positive definite matrix QQI( = ) z Solve for the matrix P form the Lyapunov equation and verify whether it is positive definite z Result: If P is positive definite, then VX ( ) < 0 and hence the origin is “asymptotically stable”. ADVANCED CONTROL SYSTEM DESIGN 23 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Lyapunov’s Indirect Theorem Let the linearized system about XXAX=Δ=Δ0 be ( ). The theorem says that if all the eigenvalues λi ()in=1,… , of the matrix A satisfy Re()λi < 0 (i.e. the linearized system is exponentially stable), then for the nonlinear system the origin is locally exponentially stable. ADVANCED CONTROL SYSTEM DESIGN 24 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Instability theorem Consider the autonomous dynamical system and assume X =0 is an equilibrium point. Let VD: → R have the following properties: ()iV (0)= 0 n (ii )∃ X0 ∈=>R ,arbitrarily close to X 0, such that V() X 0 0 (iii ) V >∀∈ 0 X U , where the set U is defined as follows UXDX=∈{: ≤ε and VX() >0} Under these conditions, X = 0 is unstable ADVANCED CONTROL SYSTEM DESIGN 25 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Summary z Motivation z Notions of Stability z Lyapunov Stability Theorems z Stability Analysis of LTI Systems z Instability Theorem z Examples ADVANCED CONTROL SYSTEM DESIGN 26 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore References z H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003. z J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991. z H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996. ADVANCED CONTROL SYSTEM DESIGN 27 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore ADVANCED CONTROL SYSTEM DESIGN 28 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore.