Introduction to Digital Systems Saint-Petersburg State University Faculty of Applied Mathematics – Control Processes Lections 8 ─ 10 prof. Evgeny I. Veremey Part 3. Synthesis of Digital Systems

1. The space X of the feasible and subset G of desirable design decisions – distance from the point x to the set G 2. Generalized design problem: 3. Generalizes necessary condition: - Frechet differential: 4. Partial corollaries from the necessary conditions: – nonlinear systems – Euler differential equations; – Riccati equations; – generalized operator equations. 1 Synthesis of Digital Systems 1. The basis of optimization approach

Discrete system model Performance objective State restrictions Purpose of control A problem of digital control systems synthesis is to find the control sequence u[n] (n=0,1,2,…) of m s -dimensional vectors from the given class, which provides achievement of the control purpose with desirable performance objectives Control restrictions 2 Synthesis of Digital Systems

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4 DLTI models of the controlled plant and controller Mathematical model of the closed system in z-domain: 2. Parametrical optimization with given admissible dynamical region Synthesis of Digital Systems

x1(t)x1(t) 0t x2(t)x2(t) x(t,γ) t1t1 t2t2 t3t3... tNtN x 5 Synthesis of Digital Systems

6 DLTI-system Desirable polynomial State controller Example: Matrix A eigenvalues The roots of desirable characteristic polynomial : Sample period: T=0.025c Synthesis of Digital Systems 3. Modal approach to the synthesis

Transient processes in the closed system for several values of the parameter γ – normalizing multiplier Equation of the closed system 7 а) output value б) control Synthesis of Digital Systems

8 DLTI-plant: Quadratic functional given on the closed system motion: State controller – The set of m s  n s -dimensional matrices with the constant components, such that all the eigenvalues of the matrix A-BK are placed in the open unit disk of the complex plane LQR-synthesis problem Synthesis of Digital Systems 4. Optimal LQR synthesis

9 The solution of the LQR-synthesis problem: Here matrix S satisfies algebraic Riccati equation: 1. Pair (A,B) is stabilizing Requirements: 2. R  0, Q>0 3. Pair (R, A-BQ -1 B T ) should not has non-stabilizing eigenvalues on the unit circle Synthesis of Digital Systems

Plant: Controller: for SISO-system 10 Synthesis of Digital Systems – max singular value 5. Synthesis on the base of matrices norms

11 Synthesis of Digital Systems Optim NCD Control μ-Tools LMI Robust Finite dimensional optimization Robust control optimization H 2 and H ∞ MATLAB synthesis support