Key Problem: determine the adjustment mechanism so that a desired performance is obtained T wo Methods: Design controller to drive plant response to mimic ideal response (error = y plant-y model => 0)Designer chooses: reference model, controller structure, Form cost function: Can chose different cost functions EX:From cost function and MIT rule, control law can be formed EX: Adaptation of feedforward gainAdjustment Mechanismy modelu y plant u c ΠΠθReference ModelPlant-+ For system where is unknown Goal: Make it look likeusing plant (note, plant model is scalar multiplied by plant)Controller: Choose cost function: Write equation for error: Calculate sensitivity derivative: Apply MIT rule: Gives block diagram:u c considered tuning parameter NOTE: MIT rule does not guarantee error convergence or stability usually kept small Based upon the concept of energy and the relation of stored energy with system stability.Example: Consider a mass spring damper system .The dynamics of this system is expressed as A corresponding state model is At any instant, the total energy V in the system consists of ◦ kinetic energy of the moving mass and◦ potential energy stored in the spring ThusThusThis means that total energy is positive unless the system is at rest at the equilibrium point xThe rate of change of energy is given by Lyapunov’s direct method makes use of a Lyapunov functionof the state may be thought as a generalized energy. When a system is described mathematically, it may not be clear what “energy” means.Consider a zero-input system described by the state equationwhere x is an n Assume that the system has only one equilibrium point The system of (1) isConsider a spherical region of radius “r ” about anequilibrium state x || x-x e || defined as follows: Let And let x −x e = An equilibrium state x e of the system of (1) is said to be stable in the sense of Lyapunov if, corresponding to eachthat trajectories starting in(ε) as t increases indefinitely. The real numberalso depends on t Choose the region An equilibrium state x e of the system of (1) is said to be asymptotically stable if it is stable in the sense of Liapunov and if every solution starting within converges, without leavingindefinitely. Note that asymptotic stability is a local concept. A knowledge of largest region of asymptotic stability is usually necessary.attraction. It is that part of the state space in which asymptotically stable trajectories originate. trajectory originating in the domain of attraction is asymptotically stable. If asymptotic stability holds for all states ( all points in the state space) from which trajectories originate, the equilibrium state is said to be asymptotically stable in the large.equilibrium statebe asymptotically stable in the large if it is stable Obviously, a necessary condition for asymptotic An equilibrium stateunstable if for some real number any real numberthere is always a state xtrajectory starting at this state leaves Theorem-1: If there exists a scalar functionfirst partial derivatives and satisfying the conditions V(x,t) is positive definite is negative definitethen the equilibrium state at the origin is uniformly If in addition, The condition of the last theorem can be stated alternately as follows: V(x,t) is positive definite is negative semidefinite Then the origin of the system is uniformly asymptotically Theorem-1: Consider a system described by If there exists a scalar functioncontinuous first partial derivatives and satisfying the conditions V(x,t) is positive definite is negative semidefinitethen the equilibrium state at the origin is uniformly Theorem-1: Consider a system described by If there exists a scalar functioncontinuous first partial derivatives and satisfying the conditions W(x,t) is positive definite in some region about is positive definite in the same region then the equilibrium state at the origin is unstable.1. The stability conditions obtained using Lyapunov functions aresufficient conditions but are not necessary conditions2. A Lyapunov function for a particular system is not unique.Thus failure to find a suitable Lyapunov function to showstability or asymptotic stability or instability of the equilibrium state under consideration can give no information on stability.3. Although a particular Lyapunov function may prove that theequilibrium state under consideration is stable orasymptotically stable in the regionequilibrium state, it does not necessarily mean that themotions are unstable outside the regionFor a stable or asymptotically stable equilibrium state, aLyapunov function with the required properties always exists. Consider the system described bywhere x is a nnonsingular matrix. A necessary and sufficient condition for the equilibrium state x=0 to be asymptotically stable in the large is that, The scalar function For the system of eqn(1), let us choose as a possible Lyapunov functionwhere P is a positive definite Hermitian matrix. (If x is real vector, then P can be chosen to be a positive definite real symmetric matrix). The time derivative ofas: Since V(x) was chosen to be positive definite, it is required that, for asymptotic stability, be negative definite. Thus we requirewhere Tand the matrix P is tested for positive definiteness. T o check if a given matrix A is asymptotically stable or not T ake an arbitrary real symmetric positive definite matrix Q ( an identity matrix Q=I Considering P as a real symmetric matrix , solve the Lyapunov equation A Check the definiteness of the matrix P If P>0 i.e. P is positive definite, The positive definiteness of an n×n matrix is tested using Sylvester’s criterion, which states that a necessary and sufficient condition for the matrix to be positive definite is that the determinants of all the successive principal minors of the matrix be positive. Consider the matrixwhere the overbar denotes the complex conjugate€Q(The matrix P is positive definite if all the successive principal are Positive, that is if,pLet A be an n×n symmetric matrix. A (k×k) submatrix of A formed by deleting (n-k) columns, say columns j,j,….j and the same n-k rows from A, jdeterminant of a (kminor of A.ExampleThere is one third order principal minor, which is det(A).Definition: Let A be an n×n symmetric matrix. The k-th order principal submatrix of A obtained by deleting the last (n-k) rows and columns of A is called a k-th order leading principal submatrix of A denoted as A. Its determinant is called the k-th order leading principal minor of A, denoted by |ALet A be an na. A is positive definite if and only if all its n-leading principal minors are strictlypositive.b. A is negative definite if and only if all its n-leading principal minors alternate inThe k-th order leading principal minor should have the same sign as (-1)c. A is positive semidefinite if and only if every principal minor of A is non negative.d. A is negative semidefinite if and only if every principal minor of odd order is non- Determine the stability of the equilibrium state of the following system: The system has only one equilibrium state at the origin. By Note that A is a real matrix and P must be a real symmetric Solving the Lyapunov equation If the matrix turns out to be positive definite, then x*Px is a Lyapunov function and the origin is asymptotically stable. Solving for p’s gives p Since P is positive definite, the origin of the system is asymptotically stable in the large. The Lyapunov function for this system isV( Determine the stability of the equilibrium state of the following system:By choosing Q=I and substituting I into the Lyapunov equation gives Note that A is a real matrix and P must be a real symmetric matrix LetSolving the Lyapunov equation01⎡ ⎣ ⎢ =−⎡ ⎣ ⎢ ⇒−。